• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Introduction
 Construction of indefinite ternary...
 Construction of t-forms with given...
 Applications to the presentation...
 Bibliography
 Biographical sketch
 Copyright














Title: On the representation of integers by indefinite ternary quadratic forms.
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00091628/00001
 Material Information
Title: On the representation of integers by indefinite ternary quadratic forms.
Series Title: On the representation of integers by indefinite ternary quadratic forms.
Physical Description: Book
Creator: Thoro, Dmitri Elias,
 Record Information
Bibliographic ID: UF00091628
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 001022711
oclc - 17957086

Downloads

This item has the following downloads:

Binder1 ( PDF )


Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    Introduction
        Page 1
        Page 2
        Page 3
    Construction of indefinite ternary quadratic forms with given invariants and leading coefficient
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
    Construction of t-forms with given invariants and leading coefficient
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
    Applications to the presentation of integers by certain t-forms
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
    Bibliography
        Page 50
    Biographical sketch
        Page 51
        Page 52
    Copyright
        Copyright
Full Text











ON THE REPRESENTATION OF INTEGERS

BY INDEFINITE TERNARY

QUADRATIC FORMS










By
DMITRI ELIAS THORO


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
August, 1958



















ACKNOWLEDGEMENTS


The writer wishes to express his appreciation

to the members of his supervisory committee for their

assistance and encouragement. In particular, acknowl-

edgement is made to Dr. E. H. Hadlock for his continued

counsel and interminable patience. The author also

wishes to express his indebtedness to his wife Charlotte,

and daughter Tanya for their support and inspiration.




















TABLE OF CONTENTS


Page


ACKOWLEDGEMENTS . . . .


CHAPTER

I. INTRODUCTION . . . . . . .

II. CONSTRUCTION OF INDEFINITE TERNARY
QUADRATIC FORMS WITH GIVEN INVARIANTS
AND LEADING COEFFICIENT . . .

III. CONSTRUCTION OF T-FORMS WITH GIVEN
INVARIANTS AND LEADING COEFFICIENT .

IV. APPLICATIONS TO THE PRESENTATION OF
INTEGERS BY CERTAIN T-FORMS . . .

BIBLIOGRAPHY . . . . . . . .


iii


15


. . .















CHAPTER I


INTRODUCTION


Associated with the ternary quadratic form

f = ax2 + by2 + cz2 + 2ryz + 2sxz + 2txy

is the determinant


a
d= t
a


t s
b r
r .


The greatest common divisor of the cofactors of the elements

of d is denoted by .n_ and A is defined by d = J- 2A.

-C. and A are invariants of f.

If (a, b, c, r, s, t) = 1, f is called a primitive

form. A T-form is an indefinite ternary quadratic form

which is not a zero form.


Notation. Let a, -n- and A be given. Then we shall write


a = 2ela, A = 2e2A', n = 2e3 n- where a, A', and-n- '


are odd; a = a'T2, A' = A"U2, L- = -11 V2, where a', A",

and-L 1 are square-free; a = a'/(a', A"), A = A"/(a', A"),

8 = (-L i, A 1); e = 0 or 1 according as el + e2 0 or 1

(mod 2), E = 0 or 1 according as e 0 or 1 (mod 2).








2



It will be understood that _- < 0 and A > 0,
but that.a, and hence a, may be either positive or negative.
Moreover, unless it is specified to the contrary, we shall

assume that a is prime to_-L and lal > 1, i.e., a has at
least one positive odd prime factor pi. Finally, we shall
assume that p and q are distinct odd primes, and that R is
a quadratic residue of a prime, N a quadratic non-residue.

The following important result is a restatement of
a theorem by Dr. E. H. Hadlock.1

Theorem 1.1. Given a = 2ela,_CL and A, with (a, -L ) = 1
and a odd. If for some C each of the following conditions
(1.1) (1.4) is satisfied, then we can construct a primitive
ternary quadratic form with invariants CL and A and leading
coefficient a.

(1.1) (C, 2aA) = 1.
(1.2) If 2el = 4, C E 3-L (mod 4); if 2el >_ g,
C 7fL (mod 8).

(1.3) (-_-n Cpi) = 1 for each prime divisor pi of a.
(1.4) (-aAjC) = 1.
By means of this theorem we first construct
indefinite ternary quadratic forms with given leading


E. H. Hadlock, "On the Construction of a Ternary
Quadratic Form," American Mathematical Monthly, Vol. 62
(1955), P. 532.












coefficients a and given invariants -n- and A. Use is made

of Dirichlet's Theorem and the Chinese Remainder Theorem.

After establishing a useful theorem on zero forms,

results are obtained for construction of T-forms with

given invariants and leading coefficient.

These results are then used to determine integers

represented by certain primitive indefinite ternary quad-

ratic forms.

In subsequent chapters, use shall be made of the

following lemma whenever necessary.

Lemma 1.0. For any a and A" (as previously defined) either

(i) a = 0 (mod A"), or
(ii) (a, q) = 1 where q is a prime divisor of A".

Proof. If A" = 1, (i) holds; hence assume A" > 1.

If (a, A") = 1, then any prime divisor q of A" > 1 is

prime to a, i.e., (ii) holds. If (a, A") = A", (i) holds.

Finally, if 1 < (a, A") = k < A" then any prime divisor q

of A"/k > 1 is prime to a; for let qlA"/k and (q, a) > 1.

Then qjA" and (q, a) = q or qla; hence qlk. Thus we may

write A" = k klq, k = k2 q, whence A" = k1 k2 q2, a

contradiction since A" is square-free.














CHAPTER II


cONSTRUCTION OF INDEFINITE TERNARY QUADRATIC FORMS
WITH GIVEN INVARIANTS AND LEADING COEFFICIENT

In this chapter we shall derive theorems on the
construction of indefinite ternary quadratic forms with
given invariants -J and A and given leading coefficients a.

Lemma 2.10. If-n- = -1, we can choose C so that conditions
(1.1) (1.4) are satisfied.
Proof. Take C = 1. (1.1), (1.3), and (1.4) are obviously
true. (1.3) holds since 1 -3 (mod 4) and 1= -7 (mod 8).
Lemma 2.20. If-r- = -2, A" = 1, e = 0, and a 3 (mod 8),
then we can choose C an odd prime satisfying (1.1) (1.4).
Proof. Since (a,-Ln ) = 1, we have el = 0; hence (1.2)
is satisfied vacuously. Consider the system of congruences
C =- 2 (mod pi),
C =- 1 (mod 4) if a + 1 (mod 8),
C = 3 (mod 4) if a = 5 (mod 8).
By the Chinese Remainder Theorem, these congruences have
a simultaneous solution C. Moreover, by Dirichlet's
Theorem, we may choose C an odd prime satisfying (1.1).
C satisfies (1.3) since (--fL Cip ) = (21p i) (Cipi) =
(21pi) (2p i) = 1. Now if a =- + 1 (mod g), (-aAIC) =












(-2e8AIC) = (-aC) = (Cla) = (21a) = 1. But if a 5
(mod 8), (-aAIC) = (-lIC) = -(Cla) = -(21a) = 1. Thus
(1.4) holds.
Lemma 2.21. If-n- = -2 and e = 1, we can choose C an odd
prime satisfying (1.1) (1.4).
Proof. As in Lemma 2.20 we can choose C an odd prime
satisfying (1.1) and the following conditions:
C EL 2 (mod p'), where p.is any prime
divisor of aM",
C =, 1 (mod. ) if a'A" = + 1 (mod g),
C E 5 (mod g) if a'A" = + 3 (mod 8).
From the first condition it follows that C satisfies (1.3).
If a'A" =_ + 1 (mod 8), (-aAIC) = (-2a'A"IC) =
(-21C) (a'A"IC) = (Cla'A") = (21a'A") = 1; if ,a'A" +3
(mod 8), (-aAIC) = (-21C) (a'A"jC) = -(CIa'A") =
-(21a'A") = 1, whence (1.4) holds.
Lemma 2.22. If-n- = -2, A" > 1, e = 0, and (a, q) = 1
where q is a prime divisor of A", then we can choose C
an odd prime satisfying (1.1) (1.4).
Proof. As before we can choose C an odd prime satisfying
(1.1) and the following conditions
C =. 1 (mod 4),
C = 2 (mod pi), where p is a prime
divisor of aA"/q.
C =. 1 (mod q) if &'A"q =: + 1 (mod $),











C = II (mod q) if a'A"q = + 3 (mod 8),
where N is a quadratic non-residue of q.
From the second condition it follows that C satisfies (1.3).
Now (-aAlC) = (-a'A"IC) = (Cla'A") = (CIa'A"/q) (Clq) =
(21a'A"/q) (Clq) = 1 since a'A"q = (a'A"/q) (q2) a'A"/q
(mod 8).
Lemma 2.23. If .L = -2, A" > 1, e = 0, a 0 (mod A"),
and xA" 4- 3 (mod t), then we can choose C an odd prime
satisfying (1.1) (1.4).
Proof. Take C an odd prime satisfying (1.1) and the
following conditions:
C E 2 (mod pi),
C E 1 (mod 4) if cxA" + 1 (nod 8),
C E 3 (mod 4) if aA" -- 5 (mod 8).
From the first condition it follows that (1.3) holds.
(-aAIC) = (-aA" C) = (-a/A" A"21C) = (-a/A"|C).
If aA" ac/A" =E + 1 (mod 8), we have (-aA[lC) = (Ca/A") =
(22a/A") = 1; if aA" E 5 (mod. ), we have (-aAJC) =
-(Cla/A") = -(2l /A") = 1.

Theorem 2,1. If (1) -_ = -1, or (2)-n. = -2 and
(i) e = 1, or
(ii) e = O, a =_ 0 (mod A"), and aA" l- 3 (mod 8), or
(iii) e = 0, A" > 1, and (a, q) = 1, where q is
a prime divisor of A",












then we can construct an indefinite ternary quadratic form
with invariants-JL and A, and leading coefficient a.
Proof. If -n- = -1 we apply Lemma 2.10 and Theorem 1.1.
Let _n- = -2. If e = 1, Lemma 2.21 is applicable.
If e = 0, A" = 1, and a f 3 (mod 8) we apply Lemma 2.20;
if e = 0, A" > 1, a =- 0 (mod A"), and aA" 1 3 (mod S)
we use Lemma 2.23. Now if e = 0, A" > 1, and a : 0 (mod A"),
then by Lemma 1.0 we have case (iii), hence Lemma 2.22
applies. Use of Theorem 1.1 in each case completes the proof.
Lemma 2.30. If I al = 1 and e < 2, we can choose C so that
(1.1) (1.4) are satisfied.
Proof. If e < 2, take C = 1; if e = 2, take C = 1 or -1
according as -n. = 3 or 1 (mod 4). Then (1.1), (1.2),
(1.4) are obviously true, and (1.3) is trivially true
since a I = 1.
Lemma 2,31. If a = 1, e > 3, and -n L + 1 (mod 8),
we can choose C so that (1.1) (1.4) are satisfied.
Proof. As in Lemma 2.30, take C = 1 or -1 according as
- r= -1 or 1 (mod 8).

Lemma 2.32. If a = 1, e >3 3, JL T 3 (mod 8), and
A" > 1, we can choose C an odd prime satisfying (1.1) (1.4).
e1
Proof. Let a = + 2 l. Take C an odd prime satisfying (1.1 )
and the following conditions,
C = 5 (mod 8),
C = 1 (mod A") if ( 2e C) = 1,












C -. N (mod A") if (C 2eIC) = -1, where N is
a quadratic non-residue of A".
Then C = 5 =_ 7 -n (mod 8), hence (1.2) holds. Now
(-aAIC) = (7 2eA"IC) = (T 2elC) (CIA") = 1, hence C
satisfies (1.4).
Lemma 2,33. If I a = 1, el > 3, I- f 5 (mod 8),
and A" > 1, we can choose C an odd prime satisfying
(1.1) (1.4).
Proof. Take C an odd prime satisfying (1.1) and the
following conditions:
C 3 (mod 8),
C = 1 (mod A") if (F 2elC) (-11A") = 1,
C N (mod A") if (2 2elC) (-lhA") = -1, where
N is a quadratic non-residue of A".
Since C = 3 7 -n- (mod 8), (1.2) holds. Writing
a = + 2el, we have (-aAIC) = ( 20A"IC) =
(T 2elC) (-l|A") (CIA") = 1, which establishes (1.4).
Lemma 2,34. If I a = 1, e > 3, -_n- = 3 (mod 8),
A" = 1, and e = O, we can choose C an odd prime satisfying
(1.1) (1.4).
Proof. Let C be an odd prime satisfying (1.1) and the
congruence C = 5 (mod 8). Then C = 7 -- (mod 8), hence
(1.2) holds. Since a = + 2el, (-aAIC) = (T 2eIl) =
(TlIC) = 1.












Lemma 2,35. If [ a I = 1, e >' 3, -.- = 5 (mod 8),
A" = 1, and e = 0, a < 0 or e = 1, a > O, then we can
choose C an odd prime satisfying (1.1) (1.4).
Proof. Let C = 3 (mod 8) be an odd prime satisfying (1.1).
As in Lemma 2.33, C also satisfies (1.2). Now (-aAIC) =
(: 2eA"IC) = (7 2eC) = 1 since a = + 2el and C -E 3 (mod 8).
Theorem 2.2. If a = + 2el and any one of the conditions
(1) e < 2,
(2) e > 3 and -r + +1 (mod 8),
1 -
(3) eI > 3, _n + 3 (mod g), and A" > 1,
(4) e > 3, A" = 1 and either
(i) -n 3 (mod 8) and e = 0 or
(ii) -_n. 5 (mod 8) and e = 0 if a < 0,
e = 1 if a> 0
holds, then we can construct an indefinite ternary quadratic
form with invariants nL- and A and leading coefficient a.
Proof. The proof of the theorem for cases (1) and (2)
follows from the application of Lemmas 2.30 and 2.31, and
Theorem 1.1. Case (3) is covered by Lemmas 2.32 and 2.33;
case (4) by Lemmas 2.34 and 2.35; the proof is completed by
use of Theorem 1.1.
Lemma 2.40. If e = O, a = 0 (mod A"), el < 2, and
(+t -- ja/A") = T 1, then we can choose C an odd prime
satisfying (1.1) (1.4).












Proof. Let C be an odd prime satisfying (1.1) and the
following conditions:
C -_CL (mod pi),
C 1 (mod 4) if (- I- la/A") = 1,
C = 3 (mod 4) if ( -n- la/A") = -1.
Since el < 2, (1.2) holds trivially. From the first
condition, C satisfies (1.3). If (-_JO- /A") = 1,
(-aAIC) = (-aA"IC) = (-a/A"IC) = (Cla/A")= (--_na/A") = 1.

If ( -CL ta/A") = -1, (-aAIC) = (-a/A"IC) =

(Jj(a/A l + 1)/2 ct/A"
(-1)(a/A" + 1)/2 (Cla/A") = (-1)/ (o L a/A") = 1
since a/A" = 1 (mod 2). Thus (1.4) holds.
Lemma 2.41. If e = 0, a 0 0 (mod A"), el > 2, and
(+. -J la/A") == + 1 according as -_n + 1 (mod 4), then
we can choose C an odd prime satisfying (1.1) 1.4).
Proof. Let C be an odd prime satisfying (1.1) and the
congruences C E _fL (mod pi), C 7_L- (mod 8). Thus
C satisfies (1.2) and (1.3). If -n._ a/A" =- 1 (mod 4),
then C 3 (mod 4) and we have (-aAIC) = (-a/A"IC) =
-(Cla/A") = -(--n- li/A") = -(-n- a/A") = 1; if 1
(mod 4) and a/A" = 3 (mod 4), then (-aAIC) = (-a/A"IC) =
(Cla/A") = (-_L a/A") = -( -L_ la/A") = 1. Finally,
if -J. -- 3 (mod 4), C f 1 (mod 4), and hence (-aAIC) =
(Cla/A") = (- -- la/A") = 1. Thus (1.4) holds.












Lemma 2,42. If e = 0, A" > 1 and (a, q) = 1 where q is
a prime divisor of A", then we can choose C an odd prime
satisfying (1.1) -(1.4).
Proof. Let C be an odd prime satisfying (1.1) and the
following conditions:
C = _.- (mod pi), where pi is a prime divisor
of aA"/q,
C = 7 -n (mod. ),
C E 1 or N (mod q) according as

(-)(-n- + 1) (a'A" + 1)/4(_ jt'"/q)

1 or -1, and where N is a quadratic non-
residue of q.
From the first condition it follows that C satisfies

(1.3) and from the second that (1.2) holds for all el
We also have (-aAIC) = (-a'A"jC) =

(-l)(C 1)(a'A" + 1)/4 (Cla'A"/q) (Cq) =

(-)(n- + 1)('A" + 1)1/4 (- la 'Al"/q) (Cq) = 1,
hence (1.4) holds.
Lemma 2,43. If e = 1 and e < 2, we can choose C an odd
1 -
prime satisfying (1.1) (1.4).
Proof. Take C an odd prime satisfying (1.1) and the
following conditions:












C -f- (mod pi) where p' is a prime divisor
of aA",
If __ -- 3 (mod 4), C =- 1 or 5 (mod. ) according
as (--L ca'A") = 1 or -1,
If -_. E 1 (mod 4), C =- 3 or 7 (mod 8) according
as (-r Ia'A") = 1 or -1.
By the first condition, (1.3) is satisfied. From the last
two conditions C satisfies (1.2) since e < 2. Moreover,
1

(-aAIC) = (-2A"IC) = (21C)(-1)(c 1)('A" + 1)/4(CIlaA,)

(21C)(-1)(-- + 1)(a'A" + 1)/4(- IccA"). Thus if
-n- = 3 (mod 4) and (- -n- ca'A") = 1, (-aAIC) =

(21C) (- -n- Ja'A") = (+1) (+1) = 1. If _- r- 1 (mod 4)
and ( -- la'A") = +1, (-aIC) =

(21C) (-1) (-l1a'A") ( -n ja'2") =
-(2C1) ( -n- la'A") = -(T1) (+1) = 1. Thus (1.4) holds.
Lemma 2.44. If e = 1, e > 3, A" > 1, and (a, q) = 1
where q is a prime divisor of A", then we can choose C an
odd prime satisfying (1.1) (1.4).
Proof. Let C be an odd prime satisfying (1.1) and the
following conditions:
C = '. (mod p'), where p' is a prime divisor
of A"/q,
of acA"/q,












C = 7 -n (mod 8),
C E 1 or N (mod q), according as

(-2 1-L )(-1) 1 + 1)(a 'A 1)/4(- "/q) =
-1 or 1, and where N is a quadratic non-residue
of q.
From the first two conditions we see that C satisfies

(1.3) and (1.2). Now (-aAIC) = (-21C) (a'A"IC) =

(-21C)(-l)(C 1)(W'A" 1)/4(Cla'") =

(-217--)(-CL)(1) + 1)(a'A" 1)/4(Cja' A"/q) (Clq) =


-(-21L) (-1)-+ 1)(A" 1)/(-"/q)(Cq)
by the last condition, hence (1.4) holds.
Lemma 2.45. If e = 1, e > 3, a = 0 (mod A"), and
(+ -j. JI/A") = + (21 -n-) according as _n = + 1 (m
then we can choose C an odd prime satisfying (1.1) -
Proof. Let C be an odd prime satisfying (1.1) and t]
congruences C = Js (mod p ), C = 7 _nr (mod 8).
Thus C satisfies (1.2) and (1.3). Now (-aAIC) =
(-2aA"IC) = (-21C) (a/A"jC) =

(-217 )(-l) (C-1)(a/A" 1)/4 C/A) =

-(-2j )(l)-+ l)(/A" 1)/4 Hence
-(-2i...1. )(-1) (-Rn |a/A") Hence


= 1




od 4),

(1.4).
he












if --. E- 1 (mod 4), (-aAIC) =

-(21 nL )(-1) (/A"'- 1)/2 (- _n IM/A) =
-(21-.L )(-lla/A")(-n_ I|a/A") = -(2t-n- )(-n- ja/A") = 1.
If J1. S 3 (mod 4), (-aAIC) = -(-21 -1 )(-_n.. Ia/A")
(21-n. )(-.-n- ta/A") = 1. Thus 1.4) holds.

Theorem 2.3. If any one of the following conditions holds,
we can construct an indefinite ternary quadratic form with
invariants-r and A and leading coefficient a.
(1) e = O, A"a, el < 2, and (+ -.n la/A") = 1.
(2) e = 0, A"ja, e >_ 2, and (+ -l. a/A") = 1
according as _rL. + 1 (mod 4).
(3) e = 0, A" > 1, and (a, q) = 1, where q is
a prime divisor of A".
(4) e = 1, e 2.
(5) e = 1, e 3, A" > 1, and (a, q) = 1, where
q is a prime divisor of A".
(6) e = 1, el > 3, A"ja, and (+ J-L a/A") = (21-n- )
according as -rL = + 1 (mod 4).
Proof. The proof follows from the applications of
Lemmas 2.40 2.45 and Theorem 1.1.

















CHAPTER III


CONSTRUCTION OF T-FORMS WITH GIVEN INVARIANTS

AND LEADING COEFFICIENT


In this chapter we first establish a useful theorem

on zero forms. Results are then obtained on the construc-

tion of T-forms with given invariants -n and A and leading

coefficient a.

Lemma 3.10. Given f = f (x, y, z)
ax2 + by2 + cz2 + 2ryz + 2sxz + 2txy with invariants _L and

A; g = g (u, v, w) = Cu2 + v2 + a Aw,2, where C is the

third coefficient of the reciprocal form F of f; aC 0.

Then f is a zero form if and only if g is a zero form.

Proof. If f is a zero form then there exist integers

xl, y1, zl, not all zero, such that f (xl, y1, 1) = 0.
Consider the integers ul = ax1 + ty + sz, v1 = Cy Rz

where J- R = st ar, and w = zl. ul, v1, w are not

all zero, for if z 0, then wl 0; if z = 0, y 1 0,

then vI 0; and if z1 = y =O, then x f 0, hence u l 0.

But aCf (xl, Y1, Zl) = g (ul' v1' w) = 0; hence g is

a zero form.












If g (ul, V,1 w1) = 0 where ul, v1, w1 are
integers not all zero, then g (aCul, aCvl, aCwl) =
aCf (xl, 1', z ) = 0 where xl = Cul tv (Rt + sC) wl,
y1 = a (vl + Rw1), and z1 = aCw1. Since aC 0, and
xl' Y1, z1 are integers not all zero, f is a zero form.

Theorem 3.1. Given the invariants -nL and A associated with
an indefinite ternary quadratic form
f = ax2 + by2 + cz2 + 2ryz + 2sxz + 2txy.
Let a = 2ela, = e2A', -t = 2e3_L', where a, A',
and- X are odd; a = a'T2, A' = A"U2, __ = lV2

where a', A", and-frl are square-free; a a'/(a', A"),
A1 = A"/(a', A"), 8 = (_-- 1 A ), e = 0 or 1 according
as e1 + e2 0 0 or 1 (mod 2), E = 0 or 1 according as
e3 =_ 0 or 1 (mod 2). Assume (a, -- ) = (C, a-n- A) = 1,
where C, the third coefficient of the reciprocal form
F of f, is an odd prime. Then
f is a zero form if and only if

-2ea 1, -2 a l1A 1 0 C/82, -2 1 C

(3.1) are quadratic residues of C-r- /8, 8, and
al A1/8, respectively.
Proof. Assume f = f (xl, Y1, Zl) = 0 where xl, Yl, z1
are not all zero. Then












(3.2) aCf = Cx22 +_ Y 22 + ajL Azl2 = 0

where x2 = ax1 + ty1 + szl, y2 = Cyl Rzl, whence

(3.3) Cn Cx 2 + y2 + aAzl2 = 0

where x2 = _L x3. Thus we have

(3.4) 2 EICx42 + y22 + 2eal 1Z22 = 0

where e3 = 2k1 or 2k1 + 1, e1 + e2 = 2k2 or 2k2 + 1,


x4 = 2klVx3,


2 = 2 k2TU (a', A") z1.


(3.4) may,


in turn, be written


(2E C L1/8) x42 + 8y32 + (2eaA1/8) z22 = 0
if e E = O, or
(3.5) (C -L1/8) x42 + 28y42 + (aiA /6) z22 = 0
if e E = 1,
and where y2 = Sy3 = 28y4.

Since _CL 1 and alA1 are square-free, the coefficients of
(3.5) are relatively prime in pairs. Then (3.1) follows
from (3.5) by application of Legendre's theorem.1


1L. E. Dickson, Introduction to the Theory of
Numbers. (Chicago: University of Chicago Press), 1929,
Theorem 91, p. 117.












If (3.1) holds, then by Legendre's theorem there
exist integers x, y, z, not all zero such that either

(2 C lC/8) x2 + 8y2 + (2ea A /) z2= 0 or

(_ 1C/8) x2 + 28y2 + (al /6) z2 = 0.

Retracing our steps, we find that g (u, v, w) =
2 2 2
Cu + -- v + a -t Aw2 is a zero form.
Application of Lemma 3.10 completes the proof of the theorem.

Theorem 3,2. Given the invariants _-- and A associated with
an indefinite ternary quadratic form
2 2 2
f = ax + by + oz + 2ryz + 2sxz + 2txy.
Assume that (C, a -- A) = 1, where C, the third coefficient
of the reciprocal form F of f, is an odd prime. If C
satisfies (1.3) and (1.4) of Theorem 1.1, then f is a
T-form if and only if one of the following conditions
holds:

(3.6) -2 aA" is a quadratic non-residue of -L 1/6,
(3.7) -21e aE al1A-nr1C/82is a quadratic non-residue
of 6,
E
(3.8) -2 _- 1C is a quadratic non-residue of A1/6.

Proof. If (3.6) holds, then for some odd prime divisor

wI of -_ 1/6, (-2eA,"jwl) = (-2eC'A"jw ) =

(-2ea1 A w ) = -1. Thus -2ea 1 is a quadratic non-residue












of C rn1 /8, and from Theorem 3.1 it follows that f is
a T-form. If (3.7) holds, again by Theorem 3.1 f can
not be a zero form. Similarly, if (3.8) holds, -2 C
is a quadratic non-residue of a A1 /8 and hence f is a
T-fora.
If no one of (3.6) (3.8) holds then -2e a",
21e -El 2 e
-21e a 1A 1 C/82, and -2 nr 1C are quadratic
residues of -rfi /8, 8, and A, /8, respectively. Remov-
ing any square factors from a and common factors from a'
and A" we have that -2e a A is a quadratic residue of

-JL 1 /8; but by (1.4), (-aAIC) = (-2ecA'lC)
(-2ea'A"IC) = (-2ea A 1C) = 1, hence -2eal 1 is a
quadratic residue of C-.1 /8. Similarly, by (1.3)
-2 _r1 C is a quadratic residue of a and hence of

al A1 /8. Therefore (3.1) holds, whence f is a zero form.
Thus if f is a T-form then one of (3.6) (3.8) must hold
for, if not, f is a zero form as just shown.
Corollary 3.1. Let CL 1 = -1 and C satisfy (1.3), (1.4)
and (C, a -L- A) = 1. Then f is a T-form if and only if
A" > 1 and for some prime factor q of A", (a, q) = 1
and (CIq) = -(2 1q).












Proof. Since _f_ 1 = -1, 8 = 1; hence by Theorem 3.2
f is a T-form if and only if (3.8) holds, i.e., 2 C is
a quadratic non-residue of A > 1. Now by (1.3),
(- L Cpi) = (-2 t- 1Clp) = (2 CIp) = 1, hence f
is a T-form if and only if A1 > 1, and for some prime
factor q of A1, (a, q) = 1 and (2 C q) = -1. We shall
now show that an equivalent set of necessary and sufficient
conditions is obtained if A1 is replaced by A". Suppose
A1 > 1 and q is a prime such that qlA1 and (a, q) = 1;
then A" = A1 (a', A") > 1, and qIA". But if A" > 1 and
q is a prime such that q|A" and (a, q) = 1, then A1 > 1,
qjA1 and (a, q) = 1. For if A" > 1 and A1 = 1, then
1 < A" = (a', A") implies A"la', contrary to (a, q) = 1.
Now if (q, (a', A"))= q, then qla', and qcl, a contra-
diction; hence if qlA", qlA1 (a', A"), or q1A since
q is prime to (a', A").

Theorem 3.3. Let q be a prime divisor of A" > 1 with
(a, q) = 1. If -fL = -1 then we can construct a T-form
with leading coefficient a and invariants -~- and A pro-
vided there exists an odd prime C satisfying the conditions
(3.9) (C, a -f- A) =1;
(3.10) If e = 2, C =1 1 (mod 4); if e > 3, C =. 1 (mod 8);
(3.11) (- -n-Clpi) = 1 for each odd prime divisor pi of a;








21



(3.12) (-aAIC) = 1;
(3.13) (2" Clq) = -1.

Proof. If e > 2, e = 0 since a is prime to -f .
Therefore n E ~1. = -1 3 (mod 4) and. = 7 (mod 8),
hence (1.2) may be written as (3.10). Thus from conditions
(3.9) (3.12) and Theorem 1.1, it follows that we can
construct an indefinite ternary quadratio form with lead-
ing coefficient a and invariants -n and A. Moreover, by
Corollary 3.1 this form is a T-form since (3.13) implies
(Cjq) = -(2 I1q).

In the following lemmaslet a be prime to q, an
odd prime divisor of A" > 1. We shall make use of the
congruences
(3.14) C 2 (mod pi) where pi is a prime divisor of
aA"/q,
(3.15) C = N or 1 (mod q) according as (2 l|q) = 1
or -1, and where N is a quadratic non-residue
of q.
We note that if-L = -1 and C is an odd prime satisfying
(3.14) and (3.15) then (--L cjp ) = (-2 _C 1CJ)
(2E C p) = 1 for each prime factor pi of a, and
(2E Clq) = -1; i.e., C satisfies (3.11) and (3.13).











Lemma 3.20. If-- = -1, e = E = O, a'A" 1 (mod 4)
and el < 2, then we can choose C an odd prime satisfying
(3.9) (3.13).

Proof. By the Chinese Remainder Theorem the system of
congruences (3.14), (3.15) and C = 3 (mod 4) has a simul-
taneous solution C. Thus as noted above (3.11) and (3.13)
are satisfied. Moreover, by Dirichlet's Theorem we can
choose C an odd prime satisfying (3.9). Since e < 2,
(3.10) holds. Finally (-aAIC) = (-2ea'A"IC) =
(-1C) (Cla'A") = -(Cca'A"/q) (Clq) =
-(2E la!A"/q) (-1) (2E 1q) = 1, i.e., C satisfies (3.12).

Lemma 3.21. If L 1 = -1, e = 0, E = 1, and a'A" = 1
(mod 4), (then we can choose C an odd prime satisfying
(3.9) (3.13).

Proof. As in Lemma 3.20 we can choose C an odd prime
satisfying (3.9), (3.14), (3.15) and one of the congruences
(3.16) C 3 1 (mod 4) if a'A" E 5 (mod 8),
(3.17) C E 3 (mod 4) if a'A" = 1 (mod 8).
Thus C satisfies (3.11) and (3.13). Since C = 1 and
a is prime to -_- -n- is even and a is odd; hence (3.10)
holds. Finally (-aAIC) = (-a'A"IC) =
(-l|C) (Cla'A"/q) (Cjq) = -(-1C) (21a'A"/q) (21q) =
-(-l1C) (21c'A") = 1, i.e., (3.12) holds.











Lemma 3.22. If -n = -1, e = 0, 6 = 1, and a'" 3
(mod 8), then we can choose C an odd prime satisfying
(3.9) (3.13).

Proof. Take C an odd prime satisfying (3.9), (3.14), and
(3.15) (and hence (3.11) and (3.13) ). As in Lemma 3.21,
(3.10) holds since E = 1. Also (-aIC) = (-2'A"|C) =
(CIa'A"/q) (CIq) = -(2jc'A") = 1, i.e., C satisfies (3.12).

Lemma 3.23. If -L 1 = -1, e = 1, E = 0, and el < 2,
then we can choose C an odd prime satisfying (3.9) -
(3.13).

Proof. Let C be an odd prime satisfying (3.9), (3.14)
and (3.15) and the congruence C E- 5 (mod S). Then C
satisfies (3.10), (3.11), and (3.13). Finally (-aAIC)
(-2a'A" C) = (-21C) (Cij'A"/q) (Cjq) =
-(-21C) (2 la'A") = 1, so that (3.12) holds.

Lemma 3.24. If -L1 -1, and e = 6 = 1, then we can
choose C an odd prime satisfying (3.9) (3.13).

Proof. Let C be an odd prime satisfying (3.9), (3.14),
(3.15) and one of the congruences
(3.1S) C = 1 (mod 8) if a'A" + 3 (-od 8),
(3.19) C E 5 (mod 8) if a'A" + 1 (mod 8).












Then as before (3.10), (3.11), and (3.13) hold. Now
(-aAjC) = (-2a'A"IC) = (-21C) (CJa'A"/q) (Clq) =
-(-21C) (21a'A"). Thus if a'A" = f 1 (mod 8),
C 5 (mod 8) by (3.19), hence (-aAIC) = 1; similarly,
by (3.18) (-aAIC) = 1 when a'A" I j 3 (mod 8).

Theorem 3,4. Let -. 1 = -1 and (a, q) = 1 for some prime
divisor q of A" > 1. If any one of the following conditions
holds, then we can construct a T-form with invariants -f_ and
A and leading coefficient a.
(1) e = E = 0, a'A" E- 1 (mod 4), and el < 2;
(2) e = 0, E = 1, a'A" 7 (mod 8);

(3) e = 1, E = 0, and el < 2;
(4) e = = 1.

Proof. The four conditions listed are covered by
Lemmas 3.20; 3.21, 3.22; 3.23; and 3.24, respectively;
i.e., in each case we can choose C an odd prime satisfying
(3.9) (3.13). Thus by Theorem 3.3 we can construct the
desired T-form.

Theorem 3.5. If -L = -1 and (a, q q') = 1 where q and
q' are distinct prime factors of A", then we can construct
a T-form with invariants -TL and A and leading coefficient a.












Proof. Take C an odd prime satisfying (3.9) and the
following congruences:
(3.20) C 2 E (mod p') where p' is any prime
factor of aA"/qq',
(3.21) C -E N or 1 (mod q) according as (2E |q) = 1
or -1, and where N is a quadratic non-
residue of q,
(3.22) C E= 1 or N' (mod q') according as
(2E la'A"/q) = -1 or 1 and where N'
is a quadratic non-residue of q',
(3.23) C 1 (mod 8).
From (3.20), (3.21), and (3.23) it is evident that C
satisfies (3.11), (3.13), and (3.10) of Theorem 3.3.
Also since C =E 1 (mod 8) we have (-2elC) = 1 for all e.
Thus by (3.20) (3.22) (-aAIC) = (-2ee'A"IC) =
(a'A"Ic) = (cla'A"/qq') (Clq) (C q')
(2 |la'A"/qq') (-1) (26 1q) (Cjq') =
-(2 Icx'A"/q') (Ciq') = 1, hence (3.12) holds. The theorem
then follows from Theorem 2.3.

Theorem 3.6. Let -f = -p, A" = 1, and (-2e'p) = -1.
If any one of the following conditions holds, then we can
construct a T-form with invariants CLT and A and leading
coefficient a.











(1) el < 2 and a' 1 (mod 4);
(2) el < 2; a' 3 (mod 4); and if e = 0,
then E = 1 and a' 3 (mod 8);

(3) el = 2 and e = 1.

Proof. We shall show that for each of the three cases
there exists an odd prime C satisfying (1.1) (1.4).
It then follows from Theorem 1.1 that we can construct
an indefinite ternary quadratic form with invariants -n-
and A and leading coefficient a. Moreover, this form
will be a T-form since (-2ea'lp) = (-2ealp) = -1 implies
that condition (3.6) of Theorem 3.2 holds. (There is no
loss of generality in assuming that (C, a-a A) = 1.)
Let C be an odd prime satisfying (1.1), the
congruence C =- 2 p (mod pi) and one of the congruences
(1'), (2'), or (3') according as we have (1), (2), or (3):
(1') C + 1 (mod 8) according as (-2elp)(2l') =
+ 1,
(2') C 1 or 5 (mod 8) according as
(21p) (2( Iat) = 1,
(3') C = 5 p (mod 8).
Since C =- 2E p (mod pi), (--.Cipi) = (-2 ~- CIp ) =
(2Eplpi) (Cpi) = 1, i.e., (1.3) holds. In cases (1)
and (2), (1.2) holds trivially; in case (3), (1.2) holds











by (3') since el = 2 implies e3 = 0, whence C E 3 n-
-- 3 _. -1 p (mod 4). It remains to show that C
satisfies (1.4) in each of the three cases. (-aAIC) =
(-2ea'A"lC) = (-2ea',C) =

(-2eIC) ()(C 1)(' )/4(Cla) =

(-2eIC) (_-)(C- 1)(a' 1)/4 (2 e'p.a').
From (-2ea'Ip) = -1, we have (a'jp) = -(-2elp) or
(pa') = -(-1)(p 1)(a' 1)/4(-2e p): thus (-aAIC) =
-((C l)(a' 1)/4(26 (p-1)(al'-l)/4(2e
-(-2 IC)(-l) (2 j)I=,(-l) (-2ejp)

or (-aAlC) = -(-l)(a'-l)(p+C-2)/4(-2elpC)(Ela').
Case (1): If a' 1 (mod 4), (-aAIC) =
-(-2elC)(-2e p)(26Ia') = 1 by (1'), for (-11C) =
(-21C) = 2 1 according as C ; + 1 (mod 8).
Case (2): If a' tE 3 (mod 4), (-aAIC) =

(-) (p+C)/2(-llpC)(2elpC) (2 Ia') = -(2elC)(2elp)(2 1a')
(p+C)/2 p+C-e.
since (-)(p+C)/2(-lpC) = (-1)p = -1. Thus if
e = 0, we have, from (2), (-aAIC) = -(21a') = 1.
If e = 1 then by (2') (-aAIC) = -(2 C)(21p)(2 2a') = 1.
Case (3): If el = 2 then e3 = E = O since (a,-n-) = 1.
Thus (-aAIC) = -(? 2elpC) according as a' + j 1 (mod 4);
hence by (3) and (3') (-aAIC) = -( 215) = 1.












Theorem 3.7. Let _- 1 = -p, and A" = p. If any one of
the following conditions holds, then wre can construct
a T-form with invariants -- and A and leading coefficient a.
(1) e < 2 and a' = p (mod 4);
(2) el < 2; a' = 3 p (mod 4); and if e = 0,
then E = 1 and a' 3 P (mod g);

(3) el = 2 and e = 1.
Proof. Let C be an odd prime satisfying (C, a-O-A) = 1,
one of the congruences
(1') C + 1 (mod S) according as
(2ej/ ,)(-2e-E 6 I ) = 7 1,
(2') C = 1 or 5 (mod 8) according as
(2E Ia)(211- 6 l p) = 7 1,

(3') C 5 p (mod 8)
according as we have (1), (2), or (3), and the congruences
(4') C 2 E p (mod pi),
(5') C 1 or N (mod p) according as
(2 a'lp) = -1 or 1 and where
N is a quadratic non-residue of p.
We shall show that C satisfies conditions (1.1) (1.4)
of Theorem 1.1 and condition (.3.7) of Theorem 3.2.
Since C is odd and (C, a-O-A) = 1, (1.1) holds.
(1.2) is trivial for cases (1) and (2), and in







29



Case (3) it follows from (3') since 3 TL 3 -L P
1
(mod 4).
From (4') we have (-- c&p.) = (-2--1 Cp ) =
(2E pip ) (Cip ) = 1, thus (1.3) holds.
By (5'), (Cip) = -(21e- 6 a'p) or
Ie- E I
(2 aia'Cip) = -1. Since L 1 = -p and A" = p, (a,--) =
(a', A") = 1 and hence A1 = 8 = p. Thus
_2 e- 1 a A-. L/82 = 21e- Ia'C, i.e., condition
(3.7) of Theorem 3.2 holds.
It remains to show that C satisfies (1.4).
Applying (4') and (5'), we obtain (-aAIC) =
(-2ea'AIIC) = (-2eaplC) =
(-2elC) (_-)(C-l)(a'p-1)/4(CIa,)(Cp) =

(-2eC)(-1)(C-l)(a'p-l)/4(2 (p a')(-1)(2 e- 6 a'L ) =

-(-2ejC)(-_l)(C-1)(a'p-1)/4(2 a,)(2 e- I tp)(-_l)(p-1)(a'-l)/4
Case (1): If a' = p (mod 4) we note that

(p-l)(a'-l)/4
(-)(-1)(-/= (-lip). Thus (-aAIC) =

-(-2eIC)(2E la')(-21e- I p) = 1 by (1'), since
(-2e IC) = + 1 according as C = 1 (mod 8).
Case (2): If a' = 3 p (mod 4), (-1) (a'-1)/ = 1.

Thus (-aAIC) = -(2ejC)(2E (a')(21e- CI p); if e = 0, E = 1
and a' -= 3 p (mod 8) by (2), hence (-aAIC) = 1;












if e = 1, (-aAIC) = -(21C))(2 E')(211- E p) = 1 from (2').
Case (3); Since e = 2, E = 0, and e = 1, (-aAC) =
-(P-21pC) according as a' =_ + p (mod. 8). Hence by (3')
(-aAIC) = -(F 215) = 1.
The theorem follows from Theorems 1.1 and 3.2
since we have shown there exists an odd prime C (prime
to aJAtA) satisfying (1.1) (1.4) and (3.7).

Theorem 3.8. Let 1 = -p, A" = q p, and
(-2ea/qlp) = -1, where q is a prime divisor of a. If
any one of the following conditions holds, then we can
construct a T-form with invariants J-_ and A and leading
coefficient a.
(1) el < 2 and a q (mod 4);
(2) el < 2; a 3 q (mod 4); and if e = 0,
then E = 1 and a = 3 q (mod 8);

(3) el = 2 and e = 1.

Proof. Let C be an odd prime satisfying (C, a-L-A) = 1,
one of the congruences
(1') C + p (mod 8) according as (2 Ea/q) = + 1,
(2') C p or 5 p (mod 8) according as
(2 la/q) = T 1,

(3') C -- 5 p (mod 9), according as we have
(1), (2), or (3), and the congruence











(4') C E 2 Cp (mod pi).
Thus as in Theorem 3.6, C satisfies (1.) (1.3).
Also (-aAIC) = (-2eaA"IC) = (-2ea/q q21C) =
(-2ea/qlC) = (-2eIC) (-1)(C-l)(a/q-l)/4(Cla/q). Thus
from (4') (-aAIC) = (-2e C)(-1)(C-l)(a/q-1)/4(2Epla/q) =

(-2eJC)(-1)(a/q-l)(p+C-2)/4(2 ja/q)(a/qlp) =

(-2ejC)()(a/q-l)(p+C-2)/4(2E6j/q) )(-)(-2ep) since
(a/qlp) = -(-2elp) by hypothesis.
Therefore
(-aAIC) = -(-2elpC)(-l)(a/q-l)(p+C-2)/4(2 a/q).
If a =- q (mod 4), then 1 = aq = a/q q2 a/q
(mod 4) and hence (-aAIC) = -(-2elpC)(2 la/q).
Similarly, if a 3q (mod 4), (-aAIC) =
-(2elpC)(2e a/q) since (-1)(P+C)/2(-llpC) =-1.
In Case (3), we have (-aAIC) = -(T21pC) according as
a = + q (mod 4). From (1') (3') we see that (-aAIC)
1 in each of the cases (1) (3); i.e., C satisfies (1.4).
It then follows from Theorem 1.1 that we can construct an
indefinite ternary quadratic form with invariants -C and A
and leading coefficient a. Moreover, this form is a T-form
since condition (3.6) of Theorem 3.2 holds by hypothesis.












Theorem 3.9. Let -_r 1 = -p, A" = q p and (d,q) = 1
where q is a prime. If (-2ea'qlp) = -1, then we can
construct a T-form with invariants -n- and A and leading
coefficient a.

Proof. Since (a, q) = 1 where q is a prime divisor of
A" > 1, it follows from Cases (3) (5) of Theorem 2.3
that we can construct an indefinite ternary quadratic
form with invariants --L and A and leading coefficient a.
There is no loss of generality in assuming that
(C, a-L A) = 1. Since (-2ea'qlp) = -1, -2eaA" is a
quadratic non-residue of-rL-1 /8 = -p; thus (3.6) of
Theorem 3.2 holds, and hence the constructed form is
a T-form.

Theorem 3.10. Let _L = -p, A" = q p, (a,q) = 1
and (-2ea'qjp) = 1 where q is a prime. If any one of
the following conditions holds, then we can construct a
T-form with invariants _-A and A and leading coefficient a.

(1) e1 < 2 and a' = q (mod 4);
(2) el < 2; a' = 3q (mod 4); and if e = 0,
then E = 1 and a' E 3q (mod 8);

(3) el = 2 and e = 1.












Proof. Let C be an odd prime satisfying (C,a-D-A) = 1,
one of the congruences
(1') C +E p (mod 8) according as
(2 aa'q) = :- 1,
(2') C E p or 5P (mod 8) according as
(2 ja'q) = + 1,
(3') c = 5p (mod 8),
according as we have (1), (2), or (3), and the congruences
(4') C = 2 p (mod pi),
(5') C 1 or N (mod q) according as
(2 pjq) = 1 and where N is a
quadratic non-residue of q.
As in Theorem 3.6, C satisfies (1.1) (1.3).

Also (-aAC) = -(-1)('q-1)(C+-2)/4 (-epC)(26 la).
Thus if a' t q (mod 4), (-aAIC) = -(-2ejpC)(2eia'q);
if a' 3q (mod 4), (-aAIC) = -(2e pC)(2la'q).
Hence (-aAIC) = -(T 2elpC)(2 E a'q) according as
a' = + q (mod 4). From (1') (3') it follows that
(-aAIC) = 1 in each of the cases (1) (3); hence C
satisfies (1.4). Thus Theorem 1.1 is applicable.
From (5') we have (C q) = -(2EpIq) or
(2 pCIq) = -1 which implies that condition (3.5) of
Theorem 3.2 holds since -L- = -p, =q, and = 1.
1 )l q ad8=1












The following theorem summarizes the preceding
results.

Theorem 3.11. Let -.L = -p, and A" = 1, p, or q, where
p and q are distinct primes. Let (A) be the condition that

(-2e 'A,' p) = -1,
and (B) be the condition that any one of

(1) el < 2 and a' M AA" (mod 4);
(2) el < 2; a' = 3A" (mod 4); and if e = 0,
then E = 1 and a' -= 3A" (mod 6);

(3) e = 2 and e = 1;
holds.

We can then construct a T-form with invariants
-.r and A and leading coefficient a provided one of the

following conditions is satisfied.

(3.24) A" = p and (B) holds,

(3.25) a 0 (mod A") and both (A) and (B) hold,
(3.26) A" = q, (q, a) = 1, and either (A) or (B) holds.

Proof. We note that if A" = p or 1, (3.24) and (3.25),
respectively, are applicable. If A" = q, (3.25) and

(3.26) are applicable according as q does or does not
divide a. Thus the theorem covers all possible cases
when -_I = -p and A" = 1, p, or q.












The condition (3.24) follows directly from
Theorem 3.7.
If A" = 1, (3.25) is obtained from Theorem 3.6.
If A" = q, where q divides a, (3.25) follows from
Theorem 3.8 since (-2es/qlp) = (-2eaqlp) =
(-2ea'A" lp) and a =- a' (mod 8).
Finally if A" = q, (q, a) = 1 then (3.26) is
obtained from Theorem 3.9 and 3.10 according as
(-2ea'qlp) = -1 or 1.

We note that Theorems 3.4, 3.5, and 3.11 cover
all determinants d such that
(1) d < 3,375 or
(2) d -0 (mod 15) and d < 9,261.
For with the exception of 15, all positive integers less
than 21 are divisible by at most one odd prime. Thus
if Ij.I < 15, or j-f I < 21 and I-I 15, then

-fl- = -1 or -p. Similarly, if A < 15, or A < 21 and
A 15, then A" = 1, p, or q, where q is a prime distinct
from p. In each case one of the Theorems 3.4, 3.5, or
3.11 will be applicable, and hence determinants
d = -f- 2 < 153 = ,375 or d < 213 = 9,26 i
(mod 15) are covered.













Theorem 3.12. In general, we can construct a T-form
with invariants -L A and leading coefficient a
provided
(1) a is prime to two or more prime divisors
of A", or
(2) one of the conditions (1) (6) of Theorem 2.3
is satisfied and for some prime divisor u of -CL 1/
we have (alu) = -(-2eA"U ).

Proof. Let (a, qq') = 1 where q, q' are distinct prime
divisors of A". We may choose C an odd prime such that
(C, a _CLA) = 1, C E 7-n- (mod 8), and C = -_n (mod pl),
where pi is any prime divisor of aA"/qq'. Thus (1.1) -

(1.3) of Theorem 1.1 are satisfied. Since (a, qq') = 1,
we are still free to choose C with respect to moduli
q and q'. Since q must divide either 8 or 1A/6, and
hence either 8 > 1 or A1/8 > 1, we may first choose
C = k (mod i) so that one of conditions (3.7), (3.S)
of Theorem 3.2 is satisfied. Thus
(-aAIC) = [(-2eij)(a'A"/qq'IC)(qjC)](q'lC) where
the expression in brackets is completely determined
by the preceding congruences. Finally, C may be chosen
(with respect to q') so that (-aAIC) = 1. Thus Theorems
1.1 and 3.2 are satisfied.








37



If one of conditions (1) (6) of Theorem 2.3
is satisfied, then we can construct an indefinite ternary
quadratic form with invariants -n-, A and leading coef-
ficient a. In the derivation of Theorem 2.3 we took C
an odd prime satisfying (13 (1.4). There is no loss
of generality in assuming that (C, a-ILA) = 1. From the
second requirement in (2) it follows that condition (3.6)
of Theorem 3.2 holds. Thus our indefinite form is a T-form.

















CHAPTER IV


APPLICATIONS TO THE REPRESENTATION OF INTEGERS
BY CERTAIN T-FORMS

We shall use the theorems of the preceding chapter
to find integers represented by the primitive T-forms
listed in Table III of Dickson's Studies in the Theory

of Numbers.
We note that each of the T-forms in this table
has a determinant

d = p2, 2p2, 22p, 22p2, 23p, 24p,

23p2, p3, 2p3, 2pq, 22pq, or p2q.


Applications of Theorem 3,6.
We shall consider applications of Theorem 3.6
to the T-forms of determinant

d = p2, 2p2, 22p2, 23p2.



L. E. Dickson, Studies in the Theory of Numbers
(Chicago: University of Chicago Press), 1930, p. 151.

39












2
Theorem 4,11. Let c = p ; -n- = p = 3, 5 or 7;
A = 1; a =. 1 or 2 (mod 4); and a = R or N (mod p)
according as p = 3, 7 or 5 and where R, N are the quad-
ratic residues and non-residues, respectively, of p.
Then every T-form of determinant p2 represents a and

hence 22ka.

Proof. Since -r-= -p and A = 1, we have-- = -p
A" = 1 and e2 = = 0. We shall show that Theorem

3.6 is applicable.
Case 1: el= 0. Then a = a, e = 0, and

(-2ea'lp) = (-2eap) = (-llp) (alp) = -1 since a = a = R
or N (mod p) according as p = 3, 7 or 5. Moreover,
a = a =_ 1 (mod 4) and hence (1) of Theorem 3.6 is
2
satisfied. Thus we can construct a T-form f = ax +
with invariants _L = -p and A = 1; but by Table III,
all T-forms of determinant d = p2 are equivalent to the
representative form and hence represent a = f (1, 0, 0).
Case 2: el = 1. Then e = 1 and a = 2a. If p = 3, 7
then a = 2a R (mod p), hence (-2alp) =

(-l1p)(2alp) = -1; if p = 5, 2a =- N (mod 5), and again
(-2alp) = -1. Finally (1) and (2) of Theorem 3.6 hold;
hence we may construct a T-form f = a:2+ ... with deter-
minant p2 and where a 2 (mod 4). Therefore as in












Case 1, every T-form of determinant d = p2 represents
a = 2 (mod 4) if a is also subject to the restrictions
indicated above.
We note also that if f(x1, Y1, z1) = a then
f(2-xl, 2k1y, 2kzl) = 22ka.

Theorem 4.12. Let d = 2p2; _n-= p = 3, 5;
A = 2; a =_ 1 (mod 2) or a 2 (mod 8); and
a = N or R (mod p) according as p = 3 or 5. Then every
T-form of determinant 2p2 represents a and hence 22ka.
Proof. .L 1 = -P, A" = 1, e3 = E = 0, and e2 = 1.
Case 1: el = 0. Then a = a, e = 1 and
(-2alp) = (-21p) (alp) = -1 since a =- N or R (mod p)
according as p = 3 or 5. Since a =_ 1 (mod 2) we also
have conditions (1) and (2) of Theorem 3.6 satisfied.
Case 2. el = 1. Then e = 0, a = 2a, and
(-alp) = (-4alp) = (-21p) (alp) = -1 as before.
If a. S 2 (mod 8) then c = 1 (nod 4) and (1) of
Theorem 3.6 is satisfied.
The theorem follows in both cases since there is
but one class of T-forms of determinant d = p2. As in
Theorem 4.11, every form representing a also represents
22ka. We shall draw this conclusion in subsequent theorems.












Theorem 4,13. Let d = 22p2 and a 1 (mod 3).
Then every T-form with invariants -r. = -6, A = 1
represents every a =- 1, 3, or 5 (mod 8), and at least
one T-form with invariants -r = -3, A = 4 represents
every a = 1 or 2 (mod 4).

Proof. If-n- = -6 and A = 1, then -L = -3, = 1,
e2 = 0, and e3 = E = 1. Since (a, -- ) = 1, we must
have el = 0. Then e = O, a = a, and
(-alp) = (-llp) (alp) = -1 since p = 3. We also have
(1) and (2) of Theorem 3.6 holding according as
a = a =- 1, 5, or 3 (mod T). Thus as in Theorem 4.12,
a is represented by every T-form with invariants
- = -6, A = 1.
If -r. = -3, A = 4, then -T1 = -3, A" = 1,
e2 = 2, and e3 = E = 0. It then follows, as in the
proof of Theorem 4.11, that we can construct a T-form
with leading coefficient a = 1 (mod 3), where a E 1
or 2 (mod 4). By Dickson's Table III, this form is
equivalent to one of two forms with invariants -0- = -3
and A = 4.

Theorem 4.14. Let d = 23p2 and a S 2 (mod 3).

Then at least one T-form with invariants -fL= -3,
A = g represents every a = 1 (mod 2) or a = 2 (mod 8),













and every T-form with invariants -- = -6, A = 2

represents every a =- 1 (mod 2).

Proof. If-L = -3, A = g, then -L_ = -3, A" = 1,

e2 = 3, e3 = 3 = 0 and the proof of Theorem 4.12

is applicable.

If -fL = -6, A = 2, then -f = -3, A" = 1 and
e2 = e3 = E = 1. Let e = 0. Then e = 1, a = a and

(-2alp) = (-2(p) (zap) = -1 since p = 3 and a = a = 2

(mod 3). Since e = 1, (1) and (2) of Theorem 3.6 hold

for any odd a.

The theorem then follows as in the proof of
Theorem 4.13.


Applications of Theorem .4.

In the following theorems we shall consider
applications of Theorem 3.4 to T-forms of determinant

d = 22p, 23p, 24p, 2pq, 22pq.

Theorem 4.21. Let d = 22p where p = 3, 5, 7, 11, 13, 19,

and (a, p) = 1. Then at least one T-form with invariants
-r- = -1, A = 22p represents every a _- 2 or p (mod 4),

and every T-form with invariants -n_ = -2, A = p represents
every a = p, 3p, or 5p (mod 8).












Proof. If-L = -1, A = 22p, then -f_ = -1, A" = p,
e2 = 2, e3 = =0.
Case 1: e = 0. Then a = a and e = 0. Since E = 0

and a = a =. A" = p (mod 4), (1) of Theorem 3.4

is satisfied.
Case 2: el = 1. Then a = 2a, e = 1, and hence

(3) of Theorem 3.4 holds for a 2 (mod 4).
If-F- = -2, A = p, then -n 1 = -1, A" = p,
e2 = 0, and e = 6 = 1. Let el = 0. Thus a = a,
e = 0, and condition (2) of Theorem 3.4 holds since
a p, p, 5p (mod g) implies that a 0 7p = 7A"

(mod g), i.e., aA" 1 7 (mod 8).

Hence by Theorem 3.4 we can construct a T-form
with the specified invariants which represents a.
When -- = -1, A = 22p, this T-form will be equivalent
to one of two entries in Table III; when = -2,

A = p, the constructed form is equivalent to the single
entry found in Table III.

Theorem 4.22. Let d = 23p, p = 3, 5, 7 and (a, p) = 1.
Then at least one T-form with invariants .-t = -1,
A = 23p represents every a =- 1 (mod 2) or a E 2p

(mod 8), and every T-form with invariants -fL = -2,
A = 2p represents every a = 1 (mod 2).













Proof. If-f= -1, A = 23p, then _. = -1, A" = p,

e2 = 3, and e = E = 0.

Case 1: el = 0. Then a = a, e = 1, whence (3)

of Theorem 3.4 is satisfied for a = 1 (mod 2).

Case 2: e = 1. Then a = 2a and e = 0. Since

a 2p (mod a), a p (mod 4) and hence (1) of

Theorem 3.4 holds.

If -.L = -2, A = 2p, then _.1 = -1, A" = p,
e2 = 1 and e3 = E = 1. For eI = we have a = a c 1

(mod 2) and e = 1, hence (4) of Theorem 3.4 holds.

The theorem then follows from Table III.

fl 4
Theorem 4.23. Let d = 2 p, p = 3, 5 and (a, p) = 1.

Then at least one T-form with invariants -n- = -1,
A = 2 p represents every a = 2, p (mod 4); at least one
T-form with invariants -n_= -2, A = 22p represents every

a =_ p, 3p or 5p (mod 8); and at least one T-form with
invariants _L = -22, A = p represents every a = p (mod 4).

Proof. The proof is similar to that of Theorem 4.21.

When-.L = -1, (1) and (3) of Theorem 3.4 are satisfied

according as el = 0, 1. Then-rL = -2 or -22 we must

have el = 0; it is readily seen that conditions (2) or

(1), respectively, of Theorem 3.4 are then satisfied.












Theorem 4,24. Let d = 2pq, p = 3 and q = 5, 7, 11, 13;
or p = 5, q = 7. Then if a is prime to either p or q,
at least one T-form with invariants -CL = -1, A = 2pq

represents every a =. 1 (mod 2) or a = 2 pq (mod 8).

Proof. We have -o 1 = -1, A" = pq, e2 = 1, and
e = 6 =0.

Case 1: el = 0. Then a = a, e = 1, and (3) of
Theorem 3.4 holds for a =- 1 (mod 2).
Case 2: el = 1. Then a = 2a, e = 0, and (1) of

Theorem 3.4 holds if a =- 2pq (mod 8).

Theorem 4.25. Let d = 22pq and a be prime to either p or q.

Then at least one T-form with invariants -n- = -1, A = 60
represents every a E 2 or 3 (mod 4), and at least one

T-form with invariants --L = -2, A = 15 represents every
a = 3, 5 or 7 (mod 8).

Proof. As in the preceding theorems, the proof follows
from Theorem 3.4.


Applications of Theorem 3.5.
The following theorem is an application of

Theorem 3.5 to T-forms of determinant d = 2pq or 22pq.












Theorem 4.3. Let d = 2pq or 22pq, where p, q are distinct

odd primes such that d < 83. Then at least one T-form of

determinant d represents every a if (a,-Lpq) = 1.

Proof. It is evident that __. = -1 and A" = pq in each

case. Also a, and hence a, is prime to pq; thus by Theorem

3.5 we can construct a T-form of determinant d which rep-
resents a. The theorem follows from Table III.


Applications of Theorems 3,4, 3,7 3.10.

We shall consider applications of these theorems

to T-forms of determinant

d = p3, 2p3, p2q.

Theorem 4.41. Let d = p3, (a, 3) = 1 and

a E- 2, 3 (mod 4). Then every T-form with invariants
-L = -1, A = 27, and at least one T-form with invariants

- = -3, A = 3 represents a.


Proof. If -n-= -1, A = 27, then _L = -1, A" = 3

and e2 = e3 = = 0. The proof then follows, as in

Theorem 4.21, from the application of Theorem 3.4.

If -L= -3, A = 3, then = -3, A" = 3,

and e2 = e3 = = 0.












Case 1: el = 0. Then a = a _. 3 (mod 4), and
(1) of Theorem 3.7 holds.
Case 2: el = 1. Then a = 2a = 2 (mod 4), e = 1,
and hence conditions (1) or (2) of Theorem 3.7 are
satisfied according as a E- 3 or 1 (mod 4).

Theorem 4.42. Let d = 2p3, (a, 3) = 1 and
a 1 (mod 2) or a = 6 (mod 8). Then every T-form with
invariants -r-= -1, A = 54, and at least one T-form with
invariants -a = -3, A = 6 represents a.

Proof. If -f = -1, A = 54, then the proof is similar to
that of Theorem 4.22.
If -fL= -3, A = 6, then-CL 1 = -, A" =3,
e2 = 1 and e3 = E = 0.
Case 1: el = 0. Then a = a 1 (mod 2), e = 1
and (1) or (2) of Theorem 3.7 holds according as
a = 3 or 1 (mod 4).
Case 2: el = 1. Then a = 2a = 6 (mod 8), e = 0,
and (1) of Theorem 3.7 holds.

Theorem 4.431. Let d = p2; = 3, q = 5, 7 or

p = 5, q = 3; and (a, q) = 1. Then every T-form with
invariants -CL= -1, A = p2q represents a 2, q (mod 4).












Proof. Since --1 = -1, A" = q, e2 = e3 = E = 0,
we have conditions (1) or (3) of Theorem 3.4 satisfied
according as a =- q or 2 (mod 4).

Theorem 4.432. Let d = p2q and (a, q) = 1.
Then a =- 2 (mod 3), a -- 1 (m9d 3), or a a -+ 1 (mod 5)
are represented by at least one T-form with invariants
-- = -p, A = q, according as d = 45, 63, or 75.

Proof. If -L= -3 and A = 5, then-l.1 = -3, A" = 5
and e2 = e = = 0. Since a = 2ela =. 2 (mod 3),
a C- 1, 2 (mod 3) according as el E 1, 0 (mod 2).
But e1 e (mod 2), hence (-2ea'qlp) =
(-2e'5a13) = (2e,13) = -1 and hence by Theorem 3.9 we
can construct a T-form with invariants -n-= -3, A = 45
and leading coefficient a.
Similarly, when -r. = -3, A = 7, or -CL = -5,
A = 3, we find that (-2eaqlp) = -1, and hence Theorem 3.9
is applicable.

Theorem 4.433. Let d = p2q, (a, q) = ,
and a =- 2, q (mod 4). Then a -= 1 (mod 3),
a =- 2 (mod 3), or a = =+ 2 (mod 5) are represented by
at least one T-form with invariants -- = -p, A = q,
according as d = 45, 63, or 75.












Proof. As in Theorem 4.432 we have _r-. = -p, A" = q
and e2 = e3 = = 0. However, in each case
(-2eta'qp) = 1. When a q (mod 4), el = e = 0, and
condition (1) of Theorem 3.10 is satisfied. When a 2 2
(mod 4), el = e = 1, a = 2a, and (1) or (2) of Theorem
3.10 hold according as a = + q (mod 4).

Theorem 4.434. Let d = p2q, a O (mod q),
and a =- 2, q (mod 4). Then a 2 (mod 3),
a 1 (mod 3), or a E 1 (mod 5) are represented by
at least one T-form with invariants l= -p, A = q
according as d = 45, 63, or 75.

Proof. If -n = -3 and A = 5, then 1 = -3, = 5,
and e2 = e3 = = 0. Since a = 2ela = 2el.5a' = 2 (mod 3),
a/5 = a' =_ 20 2e (mod 3), and hence (-2e'/qlp) =
(-2ea/513) = (-113) = -1. Moreover, when a =E q (mod 4),
el = e = 0; when a E 2 (mod 4), el = e = 1. In the former
case, (1) of Theorem 3.8 holds; in the latter, (1) or (2)
of Theorem 3.8 holds according as a E q, 3q (mod 4).
Similarly, when -L = -3, A = 7, or -rL = -5,
A = 3, Theorem 3.8 is applicable and hence we can
construct a T-form representing a.















BIBLIOGRAPHY


Dickson, L. E. Introduction to the Theory of Numbers.
Chicago: University of Chicago Press, 1929.

Modern Elementary Theory of Numbers.
Chicago: University of Chicago Press, 1939.

__ Studies in the Theory of Numbers. Chicago:
University of Chicago Press, 1930.

Jones, Burton W. The Arithmetic Theory of Quadratic
Forms. Carus Mathematical Monograph Number Ten.
Baltimore: The Mathematical Association of America,
Waverly Press, 1950.

"A New Definition of Genus for Ternary Quadratic
Forms." American Mathematical Society Transactions,
Vol. 33, 1931, pp. 92-110.

Jones, Burton W., and Hadlock, E. H. "Properly Primitive
Ternary Indefinite Quadratic Genera of More Than One
Class," Proceedings of the American Mathematical
Society, Vol. 4, August 1953, pp. 539-543.

Hadlock, E. H. "On The Construction of a Ternary Quadratic
Form;" American Mathematical Monthly, Vol. 62,
August-September 1955, p. 532.

Ross, Arnold E. "On Representation of Integers by Indefinite
Ternary Quadratic Forms of Quadratfrei Determinant,"
American Journal of Mathematics, Vol. 55, 1933,
pp. 293-302.

"On Representation of Integers by Quadratic
Forms," Proceedings of the National Academy of Sciences,
Vol. 18, 1932, pp. 600-'oi0.
















BIOGRAPHICAL SKETCH


The writer of this dissertation was born in

Yonkers, New York, February 2, 1929. He attended public

schools in Dade County, Florida, and at 17 enlisted in

the U.S. Navy, serving for three years as an electronics

technician. Subsequently he entered the University of

Florida and .received the Bachelor of Science degree with

High Honors in 1952, and the degree of Master of Science

in 1954.

For three years he was a graduate teaching

assistant, and for one year an interim instructor in

the Department of Mathematics at the University of

Florida. His industrial experience includes employ-

ment as a mathematician by the Radio Corporation of

America, Patrick Air Force Base, Florida, and the Bureau

of the Census, Washington, D. C.

During 1957-58 he held the Dudley Beaumont Memorial

Fellowship.

The writer is a member of Phi Eta Sigma, Phi Kappa

Phi, Phi Beta Kappa, the Nathematical Association of America,

and the American Mathematical Society.













This dissertation was prepared under the

direction of the chairman of the candidate's supervisory

committee and has been approved by all members of the

committee. It was submitted to the Dean of the College

of Arts and Sciences and to the Graduate Council and was

approved as partial fulfillment of the requirements for

the degree of Doctor of Philosophy.

August 9, 1958




Dean, College 6f Arts and Sciences




Dean, Graduate School


SUPERVISORY COMMITTEE:



Chairman










/ C










Internet Distribution Consent Agreement


In reference to the following dissertation:

AUTHOR: Thoro, Dmitri
TITLE: On the representation of integers by indefinite ternary quadratic forms.
(record number: 1022711)
PUBLICATION DATE: 1958


I, ADr, 1i 7 / o-0 as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
purposes via the Internet or successive technologies.

This is a non-exclusive grant of permissions for specific off-line and on-line uses for an
indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as
prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as
to the maintenance and preservation of a digital archive copy. Digitization allows the University
of Florida or its scanning vendor to generate image- and text-based versions as appropriate and
to provide and enhance access using search software.

This grant of permissions prohibits use of the digitized versions for commercial use or profit.


Signature of Copyright Holder

D n,7",; 7-T 1 o0
Printed or Typed Name of Copyright Holder/Licensee


Personal information blurred




Date of Signature

Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department
University of Florida Libraries
P.O. Box 117008
Gainesville, FL 32611-7008




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs