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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 ...
For instance, in 1938, Nils Pipping laboriously verified the conjecture up to n = 100 000. [26] With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 10 18 (and double-checked up to 4 × 10 17 ) as of 2013.
Ribenboim defines a triply palindromic prime as a prime p for which: p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime. [5] For example, p = 10 11310 + 4661664 × 10 5652 + 1, which has q = 11311 digits, and 11311 has r = 5 digits. The first (base-10) triply palindromic ...
A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird. [4] This leads to the definition of primitive weird numbers: weird numbers that are not a multiple of other weird numbers (sequence A002975 in the OEIS). Among the 1765 weird numbers less than one million, there are ...
In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the weak Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes.
Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff primes are known. 100: Centesimal: As 100=10 2, these are two decimal digits. 121: Number expressible with two undecimal digits. 125: Number expressible with three quinary digits. 128: Using as 128=2 7.
The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner's number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50 drawers in any order.
The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by Nick Trefethen . A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. The deadline for the contest was May 20, 2002.
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