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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. 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.
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. 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 Mersenne prime M p = 2 p − 1 has a corresponding perfect number M p ...
List of recreational number theory topics; Topics in cryptography; Divisibility. Composite number. ... Almost perfect number; Amicable number; Betrothed numbers;
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 ...
John Dee, wrote an occult book, which in fact was a cover for crypted text; Ibn 'Adlan: 13th-century cryptographer who made important contributions on the sample size of the frequency analysis. Duke of Mantua Francesco I Gonzaga is the one who used the earliest example of homophonic Substitution cipher in early 1400s. [2] [3]
The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form 2 p−1 M p, where M p is a Mersenne prime. [1] The perfect number 6 comes from p = 2 in this way, as 2 2−1 M 2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k , a number n is called k -perfect (or k -fold perfect) if the sum of all positive divisors of n (the divisor function , σ ( n )) is equal to kn ; a number is thus perfect if and ...
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime.For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime.