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In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. [1] 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.
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] Euclid proved c. 300 BCE that every prime expressed as M p = 2 p − 1 has a corresponding perfect number M p × (M p +1)/2 = 2 p − 1 × (2 p − 1). For example, the Mersenne prime 2 2 − 1 = 3 leads to the corresponding perfect number 2 2 ...
5 Analytic number theory: additive problems. ... List of recreational number theory topics; Topics in cryptography; ... Computational number theory is also known as ...
This is an accepted version of this page This is the latest accepted revision, reviewed on 20 January 2025. Practice and study of secure communication techniques "Secret code" redirects here. For the Aya Kamiki album, see Secret Code. "Cryptology" redirects here. For the David S. Ware album, see Cryptology (album). This article needs additional citations for verification. Please help improve ...
A major goal in cryptography is to create cryptographic primitives with provable security. In some cases, cryptographic protocols are found to have information theoretic security; the one-time pad is a common example. However, information theoretic security cannot always be achieved; in such cases, cryptographers fall back to computational ...
Some subjects generally considered to be part of analytic number theory, for example, sieve theory, [note 9] are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, [note 10] yet it does belong to analytic number theory. The following are examples of problems in analytic ...
A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert's eighth problem, and is still considered an important open problem a century later. The problem has been well-known ever since it was originally posed by Bernhard Riemann in 1860.
The quadratic nonresidue problem has both an NP and a co-NP algorithm, and so lies in the intersection of NP and co-NP. This was also true of several other problems for which zero-knowledge proofs were subsequently discovered, such as an unpublished proof system by Oded Goldreich verifying that a two-prime modulus is not a Blum integer. [29]