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Power values of certain quadratic polynomials

Anthony Flatters (2010)

Journal de Théorie des Nombres de Bordeaux

In this article we compute the q th power values of the quadratic polynomials f ∈ ℤ [ x ] with negative squarefree discriminant such that q is coprime to the class number of the splitting field of f over ℚ . The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of q which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer...

Primality test for numbers of the form ( 2 p ) 2 n + 1

Yingpu Deng, Dandan Huang (2015)

Acta Arithmetica

We describe a primality test for M = ( 2 p ) 2 n + 1 with an odd prime p and a positive integer n, which are a particular type of generalized Fermat numbers. We also present special primality criteria for all odd prime numbers p not exceeding 19. All these primality tests run in deterministic polynomial time in the input size log₂M. A special 2pth power reciprocity law is used to deduce our result.

Prime constellations in triangles with binomial coefficient congruences

Larry Ericksen (2009)

Acta Mathematica Universitatis Ostraviensis

The primality of numbers, or of a number constellation, will be determined from residue solutions in the simultaneous congruence equations for binomial coefficients found in Pascal’s triangle. A prime constellation is a set of integers containing all prime numbers. By analyzing these congruences, we can verify the primality of any number. We present different arrangements of binomial coefficient elements for Pascal’s triangle, such as by the row shift method of Mann and Shanks and especially by...

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