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Any integer greater than one is called a prime number if (and only if) its only positive divisors (factors) are one and itself. The number 1 is considered neither prime or composite but in a class of its own. It is the multiplicative identity, so it is also a unit and a divisor of unity. Every natural number is the period length of at least 1 prime. The number of factors of an integer can be found by adding 1 to the exponent of each prime factor and then calculating the product. For example, the prime factorization of 12 is 2^{2} * 3 and 12 has (2 + 1)(1 + 1) = 6 factors. The chance of a random integer x being prime is about 1/log(x). Unlike the original proof of the Prime Number Theorem from 1896 by Hadamard and Poussin, Erdös and Selberg's proof in 1949 did not employ the square root of 1. Henry Ernest Dudeney's 3by3 magic square contains "1" nonprime:
Bertrand's postulate asserts the existence of at least one prime between n and 2n. Johann H. Lambert (17281777), announced without proof that every prime number has at least one primitive root. The only proper divisor of primes. [Beedassy] Ernst Gabor Strauss (19221983) was said to have replied to a student's question about why 1 is not a prime: "The primes are the building bricks for arithmetic, and 1 is just not a brick!" Mersenne primes can be written as unbroken strings of consecutive 1s in binary form. The only number with exactly one positive divisor. [Gupta] The only number whose concatenation with itself can yield primes in many cases. [Murthy] There is only 1 "Prime Street" in England. It is in StokeonTrent, Staffordshire. [Croll] Bertrand's Postulate guarantees that in every base there is at least one prime of any given length beginning with the digit 1, and Benford's Law tells us that primes with leading digit 1 occur more often than primes beginning with any other digit in all bases. [Rupinski] Carl Sagan included the number 1 in an example of prime numbers in his book Cosmos. The smallest number n such that 10n + 1, 10n + 3, 10n + 7 and 10n + 9 are all primes. [Firoozbakht] (1) = !1, where !1 denotes subfactorial 1. [Gupta] The only number that is exactly 1/2 prime. [McAlee] If primes were called pints, then we could say, "1 is a halfpint." [McAlee] George Bernard Riemann extended Euler's Zeta function to include the sans' simple pole at s = 1. [McAlee] 1 is the only positive integer whose primal code characteristic is 1. [Awbrey] The number 1 is an "extinct" prime since it was once thought to be prime by many and now is no longer considered to be prime. [Hilliard] The remainder of division of the Mersenne numbers 2^{p}  1 by p is always equal to 1. [Capelle] The number of primes between two squares is never equal to 1. [Capelle] Henri Lebesgue (18751941) is said to be the last professional mathematician to call 1 prime. In his Elements of Algebra, Euler did not consider 1 a prime. [Waterhouse] The Egyptian fraction 1/6 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 + 1/26 + 1/33 + 1/34 + 1/35 + 1/38 + 1/39 + 1/46 + 1/51 + 1/55 + 1/57 + 1/58 + 1/62 + 1/65 + 1/69 + 1/77 + 1/82 + 1/85 + 1/86 + 1/87 + 1/91 + 1/93 + 1/95 + 1/115 + 1/119 + 1/123 + 1/133 + 1/155 + 1/187 + 1/203 + 1/209 + 1/215 + 1/221 + 1/247 + 1/265 + 1/287 + 1/299 + 1/319 + 1/323 + 1/391 + 1/689 + 1/731 + 1/901 = 1. Note that each denominator is semiprime. Found by ALLAN Wm. JOHNSON Jr. of Washington, D.C.
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