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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 ...
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. 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.
Abundant number; Almost perfect number; Amicable number; Betrothed numbers; Deficient number; Quasiperfect number; Perfect number; Sociable number; Collatz conjecture; Digit sum dynamics Additive persistence; Digital root; Digit product dynamics Multiplicative digital root; Multiplicative persistence; Lychrel number; Perfect digital invariant ...
Covers more modern material and is aimed at undergraduates covering topics such as number theory and group theory not generally covered in cryptography books. Stinson, Douglas (2005). Cryptography: Theory and Practice ISBN 1-58488-508-4. Covers topics in a textbook style but with more mathematical detail than is usual. Young, Adam L. and Moti ...
Computational hardness assumptions are of particular importance in cryptography. 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 ...
In 1988, he proposed with Hugh C. Williams a cryptographic system based on the discrete logarithmic problem in the ideal class group of imaginary-square number fields (which, according to Carl Friedrich Gauss, is related to the theory of binary-square forms), which triggered further developments in cryptography with number fields. [5]
Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist.He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least seven distinct prime factors. [1]
Silverman has published more than 100 research articles, written or coauthored six books, and edited three conference proceedings; his work has been cited more than 5000 times, by over 2000 distinct authors. [4] He has served on the editorial boards of Algebra and Number Theory and New York Journal of Mathematics. [5] [6]