Pacific Journal of
Mathematics
ON THE SOLVABILITY OF x e ≡ e (mod p)
J. B. M USKAT
Vol. 14, No. 1
May 1964
ON THE SOLVABILITY OF *« = e (mod p)
J. B. MUSKAT
1. Let e be an integer greater than 1. Let p be a prime = 1
(mod e). What conditions must p satisfy if e is an βth power residue,
modulo pi
Let βr be a fixed primitive root, modulo p. If p \ α, define ίnd a
as the least nonnegative integer t such that g* = a (mod p). For fixed
h,k,0 tί h, k tί e — 1, define the cyclotomic number (h, k) as the number
of solutions of
ind n = h (mod e), ind (n + 1) = k (mod e), 1 ^ n ^ p — 2 .
Let / = (p — 1)1 e. It is well known that
(1)
(ft, k)
(2)
(h, k) \ίϋ IV) ,
(h, k) = (k + e/2,h + e/2) ,
=
{β-h,k - A ) ,
9
k)
k)
(4)
/odd,
k = 0,
e—1
(3)
/ even ,
1/,
1,
-F
f even, h == 0 ,
f odd, h = e/2,
otherwise .
Let ζe denote a primitive e t h root of u n i t y .
e t h power c h a r a c t e r χp(a) = ζ e
THEOREM 1.
ind α
Define t h e primitive
for α ί O (mod p ) .
ind e = (p - l)/2 - Σ(λ, 0)λ (mod e).
Proof. Let z = gf (mod p). Then
β Ξ Π (1 - ^ft) (mod p) .
fc l
For a fixed v, 0 g v ^ e — 1, let Σ« a n d Πυ denote the sum and
the product, respectively, over all n, 1 ^ n ^ p — 1, such that
ind ft ΞΞ v (mod β).
Define
Then
Received April 17, 1963. This research was supported in part by the National Science
Foundation, Research Grant No. G 11309. Reproduction in whole or in part is permitted
for any purpose of the United States Government.
257
258
J. B. MUSKAT
f
χ
Set x = l.
υ
— z
= Π« (# — w) (mod p) .
Then
Ό
1 — z
== Π υ (1 — n) (mod ί?) .
Thus
e—1
ind β = Σ Σ« ind (1 — n)
= Σ Σ ί i n d ( l - Λ) - Σίind(l - u)
t;=0
Ξ
Σ ind w - ind 1 - Σί [ind (-1) + ind (n - 1)]
=
/β(β-
- Σ (Λ, 0)Λ (mod β) .
(β-l)/2
COROLLARY 1.
1/
e
is
odd,
ind
Σ
e = Σ
(mod e).
Λ[(β - fc, 0) — (fc, 0)]
2 Hereafter, let e be an odd prime.
Define the Jacobi sum
k
*U, k) = Σ Xί(n) χ P(l - n) - Σ Zί(") Zί(w + 1), i, k,j + k^O (mod β) .
It can be shown easily that
where
•B(i, v) = Σ (λ, i-vh)
.
Also, if vf is any solution of w ' = 1 (mod e),
( 5)
B(i, v) - B(i, e - v - 1) = £(w', v')
[3, p. 97] .
It will be demonstrated that for e an odd prime, ind e (mod e) can
be expressed as a linear combination of B(i, v), the rational integral
coefficients of Jacobi sums. N. C. Ankeny gave a similar criterion,
expressed in terms of the coefficients of the eth power Gaussian sum
and a variation of this criterion was found by the author [2, pp. 103, 108].
ON THE SOLVABILITY OF x* ~ e (mod p)
259
Set
(e-l)/2
S=
THEOREM
Proof.
2.
Σ
e-2
i Σ [B(e -i,v)-
B(i, v)] .
If e is an odd prime, then e ind e =Ξ S (mod e2).
If
1 ^ iS e - 1, 2 #(ΐ, v) = Σ Σ (^. * - vh)
v=l
v=l h=0
- Σ (0, i) + Σ Σ (h, i - vh)
v=l
h=\ v=l
= (β - 2) (0, i) + Σ Γ-(Λ, i) - (Λ, i + fc) + Σ (λ, * = (β - 2) (0, "0 + Σ l-(h,
^
i) - (β - Λ, ΐ) + / ] , by (1) and (4) ,
h=l
= (β - 2) (0, i) - 2[/ - (0, i)] + (β - 1)/ , by (3) ,
= (ΐ, 0)e + (β - 3)/ , by (2) .
Thus,
(6)
e[(e - ί, 0) - (i, 0)] = Σ [B(e -i,v)-
B(i, v)\ .
Substituting (6) into Corollary 1 yields the theorem.
COROLLARY 2. If e is an odd prime, e is an eth power residue,
-modulo pf if and only ifS = 0 (mod e2).
C. E. Bickmore presented without proof criteria (attributed to
L. Tanner) for e = 5 and e = 7 [1, pp. 29, 36]. These criteria, (7) and
<8), follow from Theorem 2:
Hereafter, let B(i, 1) = dif B(i, 2) = ct.
If β = 5, B(i, 3) = J5(i, 1), B(if 2) - B(3ΐ, 3) = JB(3i, 1), by (5).
5 ind 5 Ξ [2(d4 - d j + (d2 - dz)] + 2[2(d3 - d2) + (d, - d,)]
== 4(d4 - dO + 3(d3 - d2) (mod 25) .
Multiply the congruence by 6:
( 7)
5 ind 5 = dx - d4 + 7(d2 - d8) (mod 25) .
(Theorem 1 is a generalization of a proof of (7) which Emma Lehmer
communicated to the author.)
If e = 7, £(ΐ, 5) = £(ΐ, 1), £(i, 3) - E(5ί, 5) = B{U, 1), E(i, 4) - B(i, 2),
by (5). Also, B(l, 2) = J?(2, 2) - J5(4, 2), 5(3, 2) - J5(5, 2) = 5(6, 2), by
260
J. B. MUSKAT
(1) and (2).
7 ind 7 = [2(d6 - d,) + 2(c6 - cx) + (d2 - dδ)]
+ 2[2(dδ - d2) + 2(c5 - c2) + (d4 - d3)]
- d3) + 2(c4 - c3) + (d6 - d,)]
d j + 3(ώ5 - d2) + 8(cί4 - cZ3) (mod 49) .
Multiply the congruence by 39:
(8)
28 ind 7 == dx - d6 - 19(rf2 - d5) - 18(d3 - d,) (mod 49) .
The author is grateful to the referee for his helpful suggestions.
BIBLIOGRAPHY
1. C. E. Bickmore, On the Numerical Factors ofan — l (Second Notice), The Messenger
of Mathematics, 2 6 (1896), 1-38.
2. J. B. Muskat, On Certain Prime Power Congruences, Abhandlungen aus dem mathematischen Seminar der Universitat Hamburg, 26 (1963), 102-110.
3. A. L. Whiteman, The Cyclotomic Numbers of Order Ten, Proceedings of the Symposia
in Applied Mathematics, Vol. 10, 95-111, American Mathematical Society, Providence,
Rhode Island, 1960.
UNIVERSITY OF PITTSBURGH
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
ROBERT OSSERMAN
J. DUGUNDJI
Stanford University
Stanford, California
University of Southern California
Los Angeles 7, California
M. G. ARSOVE
LOWELL J. PAIGE
University of Washington
Seattle 5, Washington
University of California
Los Angeles 24, California
ASSOCIATE EDITORS
E. F. BECKENBACH
B. H. NEUMANN
F. WOLF
K. YOSIDA
SUPPORTING INSTITUTIONS
UNIVERSITY OF BRITISH COLUMBIA
CALIFORNIA INSTITUTE OF TECHNOLOGY
UNIVERSITY OF CALIFORNIA
MONTANA STATE UNIVERSITY
UNIVERSITY OF NEVADA
NEW MEXICO STATE UNIVERSITY
OREGON STATE UNIVERSITY
UNIVERSITY OF OREGON
OSAKA UNIVERSITY
UNIVERSITY OF SOUTHERN CALIFORNIA
STANFORD UNIVERSITY
UNIVERSITY OF TOKYO
UNIVERSITY OF UTAH
WASHINGTON STATE UNIVERSITY
UNIVERSITY OF WASHINGTON
*
*
*
AMERICAN MATHEMATICAL SOCIETY
CALIFORNIA RESEARCH CORPORATION
SPACE TECHNOLOGY LABORATORIES
NAVAL ORDNANCE TEST STATION
Printed in Japan by International Academic Printing Co., Ltd., Tokyo, Japan
Pacific Journal of Mathematics
Vol. 14, No. 1
May, 1964
Richard Arens, Normal form for a Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Charles Vernon Coffman, Non-linear differential equations on cones in Banach
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ralph DeMarr, Order convergence in linear topological spaces . . . . . . . . . . . . . . . . .
Peter Larkin Duren, On the spectrum of a Toeplitz operator . . . . . . . . . . . . . . . . . . . . .
Robert E. Edwards, Endomorphisms of function-spaces which leave stable all
translation-invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Erik Maurice Ellentuck, Infinite products of isols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
William James Firey, Some applications of means of convex bodies . . . . . . . . . . . . . .
Haim Gaifman, Concerning measures on Boolean algebras . . . . . . . . . . . . . . . . . . . . .
Richard Carl Gilbert, Extremal spectral functions of a symmetric operator . . . . . . . .
Ronald Lewis Graham, On finite sums of reciprocals of distinct nth powers . . . . . . .
Hwa Suk Hahn, On the relative growth of differences of partition functions . . . . . . .
Isidore Isaac Hirschman, Jr., Extreme eigen values of Toeplitz forms associated
with Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chen-jung Hsu, Remarks on certain almost product spaces . . . . . . . . . . . . . . . . . . . . .
George Seth Innis, Jr., Some reproducing kernels for the unit disk . . . . . . . . . . . . . . . .
Ronald Jacobowitz, Multiplicativity of the local Hilbert symbol. . . . . . . . . . . . . . . . . .
Paul Joseph Kelly, On some mappings related to graphs . . . . . . . . . . . . . . . . . . . . . . . .
William A. Kirk, On curvature of a metric space at a point . . . . . . . . . . . . . . . . . . . . .
G. J. Kurowski, On the convergence of semi-discrete analytic functions . . . . . . . . . .
Richard George Laatsch, Extensions of subadditive functions . . . . . . . . . . . . . . . . . . . .
V. Marić, On some properties of solutions of 1ψ + A(r 2 )X ∇ψ + C(r 2 )ψ = 0 . . .
William H. Mills, Polynomials with minimal value sets . . . . . . . . . . . . . . . . . . . . . . . . .
George James Minty, Jr., On the monotonicity of the gradient of a convex
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
George James Minty, Jr., On the solvability of nonlinear functional equations of
‘monotonic’ type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J. B. Muskat, On the solvability of x e ≡ e (mod p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zeev Nehari, On an inequality of P. R. Bessack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Raymond Moos Redheffer and Ernst Gabor Straus, Degenerate elliptic
equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abraham Robinson, On generalized limits and linear functionals . . . . . . . . . . . . . . . .
Bernard W. Roos, On a class of singular second order differential equations with a
non linear parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tôru Saitô, Ordered completely regular semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Edward Silverman, A problem of least area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robert C. Sine, Spectral decomposition of a class of operators . . . . . . . . . . . . . . . . . .
Jonathan Dean Swift, Chains and graphs of Ostrom planes . . . . . . . . . . . . . . . . . . . . .
John Griggs Thompson, 2-signalizers of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . .
Harold Widom, On the spectrum of a Toeplitz operator . . . . . . . . . . . . . . . . . . . . . . . . .
1
9
17
21
31
49
53
61
75
85
93
107
163
177
187
191
195
199
209
217
225
243
249
257
261
265
269
285
295
309
333
353
363
365