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Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.
(For related results, see Prime number theorem § Prime number race.) In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof.
The problems are posed in Gauss's Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304). [1] Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture. Gauss conjecture (class number tends to infinity)
Pages in category "Conjectures about prime numbers" The following 34 pages are in this category, out of 34 total. ... Waring's prime number conjecture;
Gauss circle problem: number theory: Carl Friedrich Gauss: 553 ... Twin prime conjecture: number theory: n/a: 1700 Ulam's packing conjecture: packing: Stanislaw Ulam:
A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written 4n + 3, with n a nonnegative integer) (sequence A002145 in the OEIS). The other prime numbers are not Gaussian primes, but each is the product of two conjugate Gaussian primes.
Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem. Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis.
One of Gauss's first results was the empirically found conjecture of 1792 – the later called prime number theorem – giving an estimation of the number of prime numbers by using the integral logarithm. [123] [o]