On cubic congruences to a prime modulus

1960 ◽  
pp. 1-7 ◽  
Author(s):  
Yu. I. Manin
Keyword(s):  
Prime Modulus

scholarly journals Pairs of additive congruences to a large prime modulus

1989 ◽  
Vol 46 (3) ◽  
pp. 438-455 ◽  
Author(s):  
O. D. Atkinson ◽  
R. J. Cook
Keyword(s):  
Trivial Solution ◽  
Prime Modulus ◽  
Image Position

AbstractThis paper is concerned with non-trivial solvability in p–adic integers, for relatively large primes p, of a pair of additive equations of degree k > 1: where the coefficients a1,…, an, b1,…, bn are rational integers.Our first theorem shows that the above equations have a non-trivial solution in p–adic integers if n > 4k and p > k6. The condition on n is best possible.The later part of the paper obtains further information for the particular case k = 5. specifically we show that when k = 5 the above equations have a non-trivial solution in p–adic integers (a) for all p > 3061 if n ≥ 21; (b) for all p execpt p = 5, 11 if n ≥ 26.

scholarly journals Dissections of a "strange" function

2015 ◽  
Vol 11 (05) ◽  
pp. 1557-1562 ◽  
Author(s):  
Scott Ahlgren ◽  
Byungchan Kim
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Generating Function ◽  
Prime Power ◽  
Partial Sums ◽  
Root Of Unity ◽  
Prime Modulus

The "strange" function of Kontsevich and Zagier is defined by [Formula: see text] This series is defined only when q is a root of unity, and provides an example of what Zagier has called a "quantum modular form". In their recent work on congruences for the Fishburn numbers ξ(n) (whose generating function is F(1 - q)), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of F(q). We prove that a more general form of their speculation is true. The congruences of Andrews–Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for ξ(n) modulo prime powers.

scholarly journals Three additive congruences to a large prime modulus

1993 ◽  
Vol 55 (3) ◽  
pp. 355-368
Author(s):  
O. D. Atkinson ◽  
J. Brüdern ◽  
R. J. Cook
Keyword(s):  
Positive Constant ◽  
Integer Coefficients ◽  
Prime Modulus ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 3 and n > 6k be positive integers. The equations, with integer coefficients, have nontrivial p-adic solutions for all p > Ck8, where C is some positive constant. Further, for values k≥ K we can take C = 1 + O(K-½).

scholarly journals Stickelberger Subideals for a Prime Modulus Related to Kummer-Type Congruences

1997 ◽  
Vol 67 (2) ◽  
pp. 203-214
Author(s):  
Takashi Agoh
Keyword(s):  
Prime Modulus

On the Factorization of Polynomials to a Prime Modulus

1935 ◽  
Vol 36 (4) ◽  
pp. 870
Author(s):  
Morgan Ward
Keyword(s):  
Prime Modulus ◽  
Factorization Of Polynomials
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