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This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2 p−1 × (2 p − 1), where 2 p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form.
The sum of Euler's totient function φ(x) over the first twenty integers is 128. [4] 128 can be expressed by a combination of its digits with mathematical operators, thus 128 = 2 8 − 1, making it a Friedman number in base 10. [5] A hepteract has 128 vertices. 128 is the only 3-digit number that is a 7th power (2 7).
Proof: 2 p+1 ≡ 2 (mod q), so 2 1 / 2 (p+1) is a square root of 2 mod q. By quadratic reciprocity, every prime modulus in which the number 2 has a square root is congruent to ±1 (mod 8). A Mersenne prime cannot be a Wieferich prime. Proof: We show if p = 2 m − 1 is a Mersenne prime, then the congruence 2 p−1 ≡ 1 (mod p 2) does ...
Perfect numbers 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, ... n is equal to the sum s ( n ) = σ ( n ) − n of the proper divisors of n .
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.
A perfect square is an element of algebraic structure that is equal to the square of another element. Square number, a perfect square integer. Entertainment
An egg that's discovered cracked in a store should not be used, as it could be dangerous.
Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 2 and can be written as 3 × 3.