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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)
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.
Category: Conjectures about prime numbers. 16 languages. ... Waring–Goldbach problem; Waring's prime number conjecture;
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.
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.
Gauss circle problem: number theory: Carl Friedrich Gauss: 553 Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane: metric geometry: Edgar Gilbert and Henry O. Pollak: Gilbreath conjecture: number theory: Norman Laurence Gilbreath: 34 Goldbach's conjecture: number theory: ⇒The ternary Goldbach conjecture, which was the ...
both are nonzero and a 2 + b 2 is a prime number (which will not be of the form 4n + 3). In other words, a Gaussian integer m is a Gaussian prime if and only if either its norm is a prime number, or m is the product of a unit (±1, ±i) and a prime number of the form 4n + 3.
An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals pZ are prime ideals of the ring Z. However, when this ideal is extended to the Gaussian integers to obtain pZ[i], it may or may not be prime. For ...