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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.
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;
(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.
Twin prime conjecture: number theory: n/a: 1700 Ulam's packing conjecture: packing: Stanislaw Ulam: Unicity conjecture for Markov numbers: number theory: Andrey Markov (by way of Markov numbers) Uniformity conjecture: diophantine geometry: n/a: Unique games conjecture: number theory: n/a: Vandiver's conjecture: number theory: Ernst Kummer and ...
The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann hypothesis.
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.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements.. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.