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A Property of Twin Primes
-
Christian Aebi
and
Grant Cairns
Published/Copyright:
August 1, 2012
Abstract.
We determine the product of the invertible quadratic residues in 
for odd n. This is a variation on Gauss' generalization of Wilson's theorem. From this we deduce that for twin primes 
, the product of the invertible quadratic residues in 
is 
, where the sign depends on the residue class of p modulo 4. We examine necessary and sufficient conditions for consecutive odd natural numbers 
to satisfy this property of twin primes. The paper concludes with two open questions.
Received: 2011-02-08
Accepted: 2011-12-19
Published Online: 2012-08-01
Published in Print: 2012-08-01
© 2012 by Walter de Gruyter Berlin Boston
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Articles in the same Issue
- Masthead
- On Weakly Complete Sequences Formed by the Greedy Algorithm
- Bijective Proofs of Vajda's Ninetieth Fibonacci Number Identity and Related Identities
- Newman Polynomials, Reducibility, and Roots on the Unit Circle
- A Property of Twin Primes
- Number of Permutations with Prescribed Up-Down Structure as a Function of Two Variables
- A -Expansion Associated to Sturmian Sequences
- On the Number of Carries Occurring in an Addition Mod
- A Probabilistic Look at Series Involving Euler's Totient Function
- Fibonacci Variations of a Conjecture of Polignac
- Extending Nathanson Heights to Arbitrary Finite Fields
- On the Maximal Cross Number of Unique Factorization Zero-Sum Sequences over a Finite Abelian Group
- An Explicit Bound for Aliquot Cycles of Repdigits
- A Note on the Minimal Number of Representations in
- A Unified Proof of Two Classical Theorems on CNS Polynomials
- On the Problem of Molluzzo for the Modulus 4
- On Multiplicative Functions with Bounded Partial Sums
- Avoiding Type or Patterns in a Partition of a Set
