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73 has 37 as the mirroring of its decimal digits. 73 is the 21st prime number, and 37 the 12th. The "mirror property" is fulfilled when 73 has a mirrored permutation of its digits (37) that remains prime. Similarly, their respective prime indices (21 and 12) in the list of prime numbers are also permutations of the same digits (1, and 2).
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 ...
Mersenne primes M p are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if 2 p − 1 is prime, then 2 p − 1 (2 p − 1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. [5] This is known as the Euclid–Euler theorem.
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
p = 2 k ± 1 or p = 4 k ± 3 for some natural number k. (OEIS: A122834) 2 p − 1 is prime (a Mersenne prime). (OEIS: A000043) (2 p + 1)/3 is prime (a Wagstaff prime). (OEIS: A000978) If p is an odd composite number, then 2 p − 1 and (2 p + 1)/3 are both composite. Therefore it is only necessary to test primes to verify the truth of the ...
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.
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In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 7 1, 9 = 3 2 and 64 = 2 6 are prime powers, while 6 = 2 × 3, 12 = 2 2 × 3 and 36 = 6 2 = 2 2 × 3 2 are not. The sequence of prime powers begins: