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4n+l A. P. INDIA a(pk Analytical Proof applying induction Bhagalpur booklet branch of Number congruence is impossible Congruences and Quadratic consecutive tm Consequences of Fermat's cyclic order Department of Mathematics elegant proofs Elementary Number Theory Fermat's and Wilson's Fermat's theorem field of Number following interesting results follows by applying form 2r+l form 4«+l form 4w Gaussian Primes integer satisfying integers Mt i=l Interesting Consequences Jacobi's reciprocity law Known Deeper Results L.E. Dickson mk+2 mk+i mk+l mv+2 number there exists odd integers Mt odd number m2 odd prime opposite parity Osmania University parity excepting preceding a prime prime number Proof for Fermat's Proof of Wilson's Quadratic Residues Dr Rameswar Rao Reciprocity Law Abstract relatively prime Results on Congruences results on Wilson's satisfy the condition satisfying the congruence Secunderabad Simple Proofs simpler theorem follows theorem of Fermat Theory of Numbers uniquely a positive Uspensky and Heaslet whenp Wilson's theorem