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*2. In particular we prove the quadratic reciprocity law and Bertrand's postulate, using fragments of 
doi:10.1016/S0304-3975(02)00617-5 
Copyright © 2002 Elsevier Science B.V. All rights reserved.
Primality test for numbers M with a large power of 5 dividing M4−1
Available online 6 February 2003.
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Abstract
The Quintic Reciprocity Law is used to produce an algorithm, that runs in polynomial time, and that determines the primality of numbers M, such that M4−1, is divisible by a power of 5 which is larger that
, provided that a small prime p, p≡1 (mod 5) is given, such that M, is not a fifth power modulo p. The same test equations are used for all such M.
A sufficiency test, together with its probability of succeeding in determining primality is given when the condition on M modulo p is omitted.
