Elementary Number Theory

a₁ asserts assume composite numbers congruence x² congruent modulo conjecture consecutive COROLLARY d₁ digits Diophantine equation Diophantus divides Division Algorithm equal Euclidean Algorithm Euler Euler's Criterion example Fermat's Theorem Fibonacci numbers finite number form 4k formula Gauss greatest common divisor hence Hint implies incongruent solutions induction infinitely many primes integers less k₁ Legendre symbol lemma linear congruence mathematician mathematics Mersenne Mersenne primes mod q multiplicative function n₁ nonresidue number theory obtain odd integer odd prime p=1 mod p₁ pairs perfect square positive divisors positive integer powers prime divisor prime factorization prime number primitive Pythagorean triple primitive root problem Proof Prove Pythagorean triangle Pythagorean triple quadratic congruence quadratic nonresidue quadratic residue relatively prime representation result satisfying sequence solvable Theorem 9-2 triangular numbers values Verify whence Wilson's Theorem x²=a mod