A Solution of the Quadratic Congruence Modulo p, p = 8n + 1, n Odd

1925 ◽  
Vol 32 (6) ◽  
pp. 294 ◽  
Author(s):  
P. A. Caris
Keyword(s):  
Congruence Modulo ◽  
Quadratic Congruence ◽  
Modulo P

A Simple Congruence modulo p

1997 ◽  
Vol 104 (5) ◽  
pp. 444-445
Author(s):  
Winfried Kohnen
Keyword(s):  
Congruence Modulo ◽  
Modulo P ◽  
Simple Congruence

scholarly journals RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

2021 ◽  
Vol 9 (VII) ◽  
pp. 2598-2601
Author(s):  
Prof. B. M. Roy
Keyword(s):  
Positive Integer ◽  
Integer Multiple ◽  
Composite Modulus ◽  
First Case ◽  
Congruence Modulo ◽  
Quadratic Congruence ◽  
First Time

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

A Simple Congruence modulo p

1997 ◽  
Vol 104 (5) ◽  
pp. 444 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Congruence Modulo ◽  
Modulo P ◽  
Simple Congruence

Connection between Fermat quotients and Euler numbers

2008 ◽  
Vol 58 (1) ◽  
Author(s):  
Stanislav Jakubec
Keyword(s):  
Euler Numbers ◽  
Fermat Quotients ◽  
Congruence Modulo ◽  
Modulo P

AbstractIn the paper, we obtain a congruence modulo p 3 among Euler numbers E p−1, E 2p−2, and Fermat quotients Q 2, Q a where p = a 2 + 4b 2.

Decomposition of congruences involving a map Φ

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
František Marko
Keyword(s):  
Arbitrary Power ◽  
Congruence Modulo ◽  
Modulo P

AbstractCongruences of Ankeny-Artin-Chowla type modulo p 2 for a cyclic subfield K of prime conductor p were derived by Jakubec and expressed in terms of a technically defined map Φ. Later, Jakubec and Lassak found a decomposition of the map Φ modulo p 2 and simplified the formulation of these congruences. A corresponding decomposition of the map Φ modulo p 3 was obtained in [MARKO, F.: Towards Ankeny-Artin-Chowla type congruence modulo p 3, Ann. Math. Sil. 20 (2006), 31–55]. That technical step was important for the formulation of congruences of Ankeny-Artin-Chowla type modulo p 3. This paper will show how to produce an analogous decomposition of the map Φ modulo an arbitrary power p n which would allow a description of analogous congruences modulo p n.

LINEAR CONGRUENCES AND RELATIONS ON SPACES OF CUSP FORMS

2007 ◽  
Vol 03 (04) ◽  
pp. 529-539 ◽  
Author(s):  
AHMAD EL-GUINDY
Keyword(s):  
Prime Ideal ◽  
Cusp Form ◽  
Modular Forms ◽  
Cusp Forms ◽  
Linear Combinations ◽  
Congruence Modulo ◽  
Hecke Eigenform ◽  
Modulo P ◽  
Linear Congruences ◽  
Level 1

Let p be a prime and let f be any cusp form of level l ∈ {2,3,5,7,13} whose weight satisfy a certain congruence modulo (p-1). Then we exhibit explicit linear combinations of the coefficients of f that must be divisible by p. For a normalized Hecke eigenform, this translates (under mild restrictions) into the pth coefficient itself being divisible by a prime ideal above p in the ring generated by the coefficients of f. This provides many instances of the so-called non-ordinary primes. We also discuss linear relations satisfied universally on the space of modular forms of these levels. These results extend recent work of Choie, Kohnen and Ono in the level 1 case.

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