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It is currently an open problem whether there are infinitely many Mersenne primes and even perfect numbers. [2] [6] The density of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given x is (e γ / log 2) × log log x, where e is Euler's ...
A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory or computer assistance.
For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6, 28, 496 and 8128. [2] The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum.
It can be proven that: . For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p + 1)-perfect.This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
Cuisenaire rods: 5 (yellow) cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green) can be evenly divided in 2 by 3 (lime green). In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. [1]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]
Noting [16] that it is easier to state for −3 and +5 than it is for +3 or −5, he states [17] the general theorem in the form: If p is a prime of the form 4 n + 1 then p , but if p is of the form 4 n + 3 then − p , is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p .