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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 ...
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. The first four perfect numbers are 6, 28, 496 and 8128. [1] The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum.
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.
When a triple of numbers a, b and c forms a primitive Pythagorean triple, then (c minus the even leg) and one-half of (c minus the odd leg) are both perfect squares; however this is not a sufficient condition, as the numbers {1, 8, 9} pass the perfect squares test but are not a Pythagorean triple since 1 2 + 8 2 ≠ 9 2. At most one of a, b, c ...
A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
This means that there should on average be about ≈ 5.92 primes p of a given number of decimal digits such that is prime. The conjecture is fairly accurate for the first 40 Mersenne primes, but between 2 20,000,000 and 2 85,000,000 there are at least 12, [ 8 ] rather than the expected number which is around 3.7.
By definition, any four-digit perfect digital invariant for , with natural number digits <, <, <, < has to satisfy the quartic Diophantine equation + + + = + + +. d 3 {\displaystyle d_{3}} has to be equal to 0, 1, 2 for any b > 3 {\displaystyle b>3} , because the maximum value n {\displaystyle n} can take is n = ( 3 − 2 ) 3 + 3 ( b − 1 ) 3 ...
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number [1]) is a natural number in a given number base that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, because = + + +.