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if the last digit of a number is 2 or 8, its square ends in an even digit followed by a 4; if the last digit of a number is 3 or 7, its square ends in an even digit followed by a 9; if the last digit of a number is 4 or 6, its square ends in an odd digit followed by a 6; and; if the last digit of a number is 5, its square ends in 25.
Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9. [55] [56] Therefore, in particular the digital root of every even perfect number other than 6 is 1. The only square-free perfect number is 6. [57]
Note that if n 2 is the closest perfect square to the desired square x and d = x - n 2 is their difference, it is more convenient to express this approximation in the form of mixed fraction as . Thus, in the previous example, the square root of 15 is 4 − 1 8 . {\displaystyle 4{\tfrac {-1}{8}}.}
Since this is also a multiple of 4 for k > 0, 2 4k ±1 ≡ ±12 (mod 20). Thus, all Mersenne numbers M 4k +1 are congruent to 11 modulo 20 and end in 11, 31, 51, 71 or 91, while Mersenne numbers M 4k −1 ≡ 7 (mod 20) and end in 07, 27, 47, 67, or 87. For the perfect numbers, define P n = 2 n−1 M n be the value which is perfect if M n is prime.
According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive powerful numbers such as (23 3, 2 3 3 2 13 2) in which neither number in the pair is a square. Walker (1976) showed that there are indeed infinitely many such pairs by showing that 3 3 c 2 + 1 = 7 3 d 2 has infinitely many solutions.
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 ...
Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3 k p where p is prime ...
All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4. For numbers with three identical digits and a fourth digit that is one higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading zero; for example: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 ...