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Displaying similar documents to “A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

A note on the diophantine equation x 2 + b Y = c z

Maohua Le (2006)

Czechoslovak Mathematical Journal

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Let a , b , c , r be positive integers such that a 2 + b 2 = c r , min ( a , b , c , r ) > 1 , gcd ( a , b ) = 1 , a is even and r is odd. In this paper we prove that if b 3 ( m o d 4 ) and either b or c is an odd prime power, then the equation x 2 + b y = c z has only the positive integer solution ( x , y , z ) = ( a , 2 , r ) with min ( y , z ) > 1 .

A note on the congruence n p k m p k n m ( mod p r )

Romeo Meštrović (2012)

Czechoslovak Mathematical Journal

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In the paper we discuss the following type congruences: n p k m p k m n ( mod p r ) , where p is a prime, n , m , k and r are various positive integers with n m 1 , k 1 and r 1 . Given positive integers k and r , denote by W ( k , r ) the set of all primes p such that the above congruence holds for every pair of integers n m 1 . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W ( k , r ) and inclusion relations between them for various values k and r . In particular, we prove that W ( k + i , r ) = W ( k - 1 , r ) for all k 2 , i 0 and...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers β , . . . , β l have the property that for each prime number p there exists a quadratic residue β j mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

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Let a d - 1 , , a 0 , where d and a 0 0 , and let X = ( x n ) n = 1 be a sequence of integers given by the linear recurrence x n + d = a d - 1 x n + d - 1 + + a 0 x n for n = 1 , 2 , 3 , . We show that there are a prime number p and d integers x 1 , , x d such that no element of the sequence X = ( x n ) n = 1 defined by the above linear recurrence is divisible by p . Furthermore, for any nonnegative integer s there is a prime number p 3 and d integers x 1 , , x d such that every element of the sequence X = ( x n ) n = 1 defined as above modulo p belongs to the set { s + 1 , s + 2 , , p - s - 1 } .

Products of factorials modulo p

Florian Luca, Pantelimon Stănică (2003)

Colloquium Mathematicae

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We show that if p ≠ 5 is a prime, then the numbers 1 / p ( p m , . . . , m t ) | t 1 , m i 0 f o r i = 1 , . . . , t a n d i = 1 t m i = p cover all the nonzero residue classes modulo p.

The largest prime factor of X³ + 2

A. J. Irving (2015)

Acta Arithmetica

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Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than X 1 + 10 - 52 .