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Landau's problems [2] 4: 4: Edmund Landau: 1912 Taniyama's problems [3] 36 – Yutaka Taniyama: 1955 Thurston's 24 questions [4] [5] 24 – William Thurston: 1982 Smale's problems: 18: 14: Stephen Smale: 1998 Millennium Prize Problems: 7: 6 [6] Clay Mathematics Institute: 2000 Simon problems: 15 < 12 [7] [8] Barry Simon: 2000 Unsolved Problems ...
The first 3 powers of 2 with all but last digit odd is 2 4 = 16, 2 5 = 32 and 2 9 = 512. The next such power of 2 of form 2 n should have n of at least 6 digits. The only powers of 2 with all digits distinct are 2 0 = 1 to 2 15 = 32 768 , 2 20 = 1 048 576 and 2 29 = 536 870 912 .
For this special case of m = 1, some of the known solutions satisfying the proposed constraint with n ≤ k, where terms are positive integers, hence giving a partition of a power into like powers, are: [3] k = 3 3 3 + 4 3 + 5 3 = 6 3 k = 4 95800 4 + 217519 4 + 414560 4 = 422481 4 (Roger Frye, 1988) 30 4 + 120 4 + 272 4 + 315 4 = 353 4 (R ...
2 35 *3 4 *5*2801*2206499*62368028479 This table suggests that the power of 2 is growing superlinearly. The best current result is that R n {\displaystyle R_{n}} is always divisible by f !, where f is about n /2.
To compute the largest power of 2 dividing the binomial coefficient () write m = 3 and n − m = 7 in base p = 2 as 3 = 11 2 and 7 = 111 2.Carrying out the addition 11 2 + 111 2 = 1010 2 in base 2 requires three carries:
[1] [2] The integers 2 3 and 3 2 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers.
For example, a 32-bit word consisting of 4 bytes can represent 2 32 distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to 2 32 − 1, or as the range of signed numbers between −2 31 and 2 31 − 1.
the even perfect numbers 2 n − 1 (2 n − 1) formed by the product of a Mersenne prime 2 n − 1 with half the nearest power of two, and; the products 2 n − 1 (2 n + 1) of a Fermat prime 2 n + 1 with half the nearest power of two. (sequence A068195 in the OEIS).