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64 is the number of codons in the RNA codon table of the genetic code. 64 is the size in bits of certain data types in some computer programming languages, where a 64-bit integer can represent values up to 2 64 = 18,446,744,073,709,551,616. Base 64 is used in Base64 encoding, and other data compression formats.
A square has even multiplicity for all prime factors (it is of the form a 2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS ). A cube has all multiplicities divisible by 3 (it is of the form a 3 for some a ).
The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×10 18. [2] That means 95,676,260,903,887,607 primes [3] (nearly 10 17), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes.
A highly composite number is a positive integer that has more divisors than all smaller positive integers. ... factors d(n) primorial ... 64 21 10080 5,2,1,1 9 ...
For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached. A037274
The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002) If n n + 1 is prime, there exists an integer m such that n = 2 2 m. The equation n n + 1 = F (2 m +m) holds in that case. [13] [14] Let the largest prime factor of the Fermat number F n be P(F n). Then,
However, in this case, there is some fortuitous cancellation between the two factors of P n modulo 25, resulting in P 4k −1 ≡ 3 (mod 25). Combined with the fact that P 4k −1 is a multiple of 8 whenever k > 1, we have P 4k −1 ≡ 128 (mod 200) and ends in 128, 328, 528, 728 or 928.
The CPU time spent on finding these factors amounted to approximately 900 core-years on a 2.1 GHz Intel Xeon Gold 6130 CPU. Compared to the factorization of RSA-768, the authors estimate that better algorithms sped their calculations by a factor of 3–4 and faster computers sped their calculation by a factor of 1.25–1.67.