Prime distribution, Legendre symbol, quadratic reciprocity, arithmetic functions, Diophantine equations and primitive roots.
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- What does the Prime Number Theorem state about the distribution of primes?
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The number of primes less than or equal to x is approximately x / ln(x).- What is the asymptotic density of primes among all natural numbers?
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Zero; lim(n -> infinity) pi(n)/n = 0.- State Bertrand's Postulate.
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For any n > 1, there exists a prime p with n < p < 2n.- What is the Twin Prime Conjecture?
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There are infinitely many pairs of primes (p, p+2) where both are prime.- What does Dirichlet's Theorem on Primes in Arithmetic Progressions state?
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If gcd(a, d) = 1, there are infinitely many primes p with p ≡ a (mod d).- What is a prime gap?
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The difference between consecutive primes; if p and q are consecutive primes, the gap is q - p.- What is the Chebyshev function psi(x)?
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The sum of log(p) over all prime powers p^k less than or equal to x.- Define the Legendre symbol (a/p) for odd prime p and integer a.
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1 if a is a quadratic residue mod p, -1 if a is a non-residue, 0 if p divides a.- State Euler's Criterion for the Legendre symbol.
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(a/p) ≡ a^((p-1)/2) (mod p).- What is the multiplicative property of the Legendre symbol?
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(ab/p) = (a/p)(b/p).- What is ((-1)/p), the Legendre symbol of -1?
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1 if p ≡ 1 (mod 4), and -1 if p ≡ 3 (mod 4).- What is ((2)/p), the Legendre symbol of 2?
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1 if p ≡ 1 or 7 (mod 8), and -1 if p ≡ 3 or 5 (mod 8).- How can you compute the Legendre symbol (a/p) efficiently?
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Using Euler's Criterion a^((p-1)/2) mod p, or by applying quadratic reciprocity repeatedly.- How does the Legendre symbol (a/p) relate to the Jacobi symbol when p is prime?
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They are equal; the Jacobi symbol generalizes the Legendre symbol to composite moduli.- State the Law of Quadratic Reciprocity for distinct odd primes p and q.
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(p/q)(q/p) = (-1)^((p-1)(q-1)/4).
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