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 TFORMS WITH GIVEN
INVARIANTS AND LEADING COEFFICIENT .
IV. APPLICATIONS TO THE PRESENTATION OF
INTEGERS BY CERTAIN TFORMS . . .
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 Tform 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', andn '
are odd; a = a'T2, A' = A"U2, L = 11 V2, where a', A",
andL 1 are squarefree; 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 nonresidue.
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 3L (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 Tforms 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 squarefree.
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. Ifn = 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. Ifr = 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. Ifn = 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. Ifn = 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 nonresidue 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 invariantsJL 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 nonresidue 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 nonresidue of A".
Since C = 3 7 n (mod 8), (1.2) holds. Writing
a = + 2el, we have (aAIC) = ( 20A"IC) =
(T 2elC) (lA") (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 1L )(1) 1 + 1)(a 'A 1)/4( "/q) =
1 or 1, and where N is a quadratic nonresidue
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) (C1)(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_ Ia/A") = (2tn )(n ja/A") = 1.
If J1. S 3 (mod 4), (aAIC) = (21 1 )(_n.. Ia/A")
(21n. )(.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
invariantsr 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 (+ JL a/A") = (21n )
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 TFORMS 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 Tforms 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", andfrl are squarefree; 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, an 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 Cr /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 squarefree, 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
Tform if and only if one of the following conditions
holds:
(3.6) 2 aA" is a quadratic nonresidue of L 1/6,
(3.7) 21e aE al1Anr1C/82is a quadratic nonresidue
of 6,
E
(3.8) 2 _ 1C is a quadratic nonresidue 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 nonresidue
of C rn1 /8, and from Theorem 3.1 it follows that f is
a Tform. 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 nonresidue of a A1 /8 and hence f is a
Tfora.
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 Tform 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 Tform 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 Tform if and only if (3.8) holds, i.e., 2 C is
a quadratic nonresidue of A > 1. Now by (1.3),
( L Cpi) = (2 t 1Clp) = (2 CIp) = 1, hence f
is a Tform 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 qA" 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 Tform
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) ( nClpi) = 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 Tform 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 lq) = 1
or 1, and where N is a quadratic nonresidue
of q.
We note that ifL = 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) =
(lC) (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 Tform 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 Tform.
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 Tform 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 nonresidue 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 Tform 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 Tform 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, aa 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 (p1)(al'l)/4(2e
(2 IC)(l) (2 j)I=,(l) (2ejp)
or (aAlC) = (l)(a'l)(p+C2)/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+Ce.
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 Tform 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, aOA) = 1,
one of the congruences
(1') C + 1 (mod S) according as
(2ej/ ,)(2eE 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 nonresidue 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, aOA) = 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.) = (21 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) (_)(Cl)(a'p1)/4(CIa,)(Cp) =
(2eC)(1)(Cl)(a'pl)/4(2 (p a')(1)(2 e 6 a'L ) =
(2ejC)(_l)(C1)(a'p1)/4(2 a,)(2 e I tp)(_l)(p1)(a'l)/4
Case (1): If a' = p (mod 4) we note that
(pl)(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) =
(P21pC) 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 Tform 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, aLA) = 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)(Cl)(a/ql)/4(Cla/q). Thus
from (4') (aAIC) = (2e C)(1)(Cl)(a/q1)/4(2Epla/q) =
(2eJC)(1)(a/ql)(p+C2)/4(2 ja/q)(a/qlp) =
(2ejC)()(a/ql)(p+C2)/4(2E6j/q) )()(2ep) since
(a/qlp) = (2elp) by hypothesis.
Therefore
(aAIC) = (2elpC)(l)(a/ql)(p+C2)/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 Tform
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 Tform 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, aL A) = 1. Since (2ea'qlp) = 1, 2eaA" is a
quadratic nonresidue ofrL1 /8 = p; thus (3.6) of
Theorem 3.2 holds, and hence the constructed form is
a Tform.
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
Tform 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,aDA) = 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 nonresidue of q.
As in Theorem 3.6, C satisfies (1.1) (1.3).
Also (aAC) = (1)('q1)(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 Tform 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 jf I < 21 and II 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 Tform
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 7n (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, aILA) = 1. From the
second requirement in (2) it follows that condition (3.6)
of Theorem 3.2 holds. Thus our indefinite form is a Tform.
CHAPTER IV
APPLICATIONS TO THE REPRESENTATION OF INTEGERS
BY CERTAIN TFORMS
We shall use the theorems of the preceding chapter
to find integers represented by the primitive Tforms
listed in Table III of Dickson's Studies in the Theory
of Numbers.
We note that each of the Tforms 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 Tforms 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 nonresidues, respectively, of p.
Then every Tform 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 Tform f = ax +
with invariants _L = p and A = 1; but by Table III,
all Tforms 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 Tform f = a:2+ ... with deter
minant p2 and where a 2 (mod 4). Therefore as in
Case 1, every Tform 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(2xl, 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
Tform 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 Tforms 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 Tform with invariants r. = 6, A = 1
represents every a = 1, 3, or 5 (mod 8), and at least
one Tform with invariants r = 3, A = 4 represents
every a = 1 or 2 (mod 4).
Proof. Ifn = 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 Tform 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 Tform
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 Tform with invariants fL= 3,
A = g represents every a = 1 (mod 2) or a = 2 (mod 8),
and every Tform with invariants  = 6, A = 2
represents every a = 1 (mod 2).
Proof. IfL = 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 Tforms 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 Tform with invariants
r = 1, A = 22p represents every a _ 2 or p (mod 4),
and every Tform with invariants n_ = 2, A = p represents
every a = p, 3p, or 5p (mod 8).
Proof. IfL = 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).
IfF = 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 Tform
with the specified invariants which represents a.
When  = 1, A = 22p, this Tform 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 Tform with invariants .t = 1,
A = 23p represents every a = 1 (mod 2) or a E 2p
(mod 8), and every Tform with invariants fL = 2,
A = 2p represents every a = 1 (mod 2).
Proof. Iff= 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 Tform with invariants n = 1,
A = 2 p represents every a = 2, p (mod 4); at least one
Tform with invariants n_= 2, A = 22p represents every
a =_ p, 3p or 5p (mod 8); and at least one Tform 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. ThenrL = 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 Tform 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 Tform with invariants n = 1, A = 60
represents every a E 2 or 3 (mod 4), and at least one
Tform 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 Tforms 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 Tform 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 Tform 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 Tforms of determinant
d = p3, 2p3, p2q.
Theorem 4.41. Let d = p3, (a, 3) = 1 and
a E 2, 3 (mod 4). Then every Tform with invariants
L = 1, A = 27, and at least one Tform 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 Tform with
invariants r= 1, A = 54, and at least one Tform 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, thenCL 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 Tform 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 Tform with invariants
 = p, A = q, according as d = 45, 63, or 75.
Proof. If L= 3 and A = 5, thenl.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 Tform 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 Tform 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 Tform 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 Tform 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. 92110.
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. 539543.
Hadlock, E. H. "On The Construction of a Ternary Quadratic
Form;" American Mathematical Monthly, Vol. 62,
AugustSeptember 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. 293302.
"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 195758 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
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