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An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. [14] All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. [15]
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.
It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, [18] thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question". [19]
The smallest odd abundant number is 945. The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in the OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. [1]
The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added.
No odd perfect numbers are known; hence, all known perfect numbers are triangular. For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
Infinitely many weird numbers exist. [3] For example, 70p is weird for all primes p ≥ 149. In fact, the set of weird numbers has positive asymptotic density. [4] It is not known if any odd weird numbers exist. If so, they must be greater than 10 21. [5] Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2 k, and