A perfect power is an integer *n* which is expressible as for integers *x* and *y* and *y* > 1.

The perfect powers up to 10000 are

- 0, 1, 4, 8, 9, 14, 21, 23, 28, 30, 41, 54, 69, 84, X1, X5, X8, 100, 121, 144, 160, 169, 183, 194, 201, 230, 247, 261, 294, 309, 344, 368, 381, 400, 441, 484, 509, 554, 5X1, 630, 681, 6E4, 714, 769, 804, 861, 900, 92E, 961, X04, X69, E14, E81, 1000, 1030, 10X1, 1154, 1209, 1228, 1284, 1323, 1331, 1341, 1400, 1481, 1544, 1609, 1694, 1708, 1761, 1830, 1901, 1985, 1994, 1X69, 1E44, 1E53, 2021, 2100, 21X1, 2284, 2369, 2454, 2541, 2630, 2721, 2814, 2909, 2X04, 2X15, 2E01, 3000, 3101, 3204, 3309, 3414, 3460, 3521, 3630, 3741, 3854, 3969, 3X84, 3E77, 3EX1, 4100, 4221, 4344, 4469, 4594, 4600, 4701, 4768, 4830, 48X8, 4961, 4X94, 5009, 5144, 5281, 5400, 5439, 5541, 5684, 5809, 5954, 5XX1, 6030, 6181, 61E4, 6314, 6469, 6604, 6761, 6900, 6X61, 7004, 705E, 7169, 7314, 7481, 7630, 77X1, 7954, 7E09, 8000, 8084, 8241, 8400, 8581, 8744, 8909, 8X94, 9061, 9230, 9401, 9594, 9769, 9887, 9944, 9E21, X100, X208, X2X1, X484, X669, X854, XX41, E030, E221, E414, E483, E609, E804, EX01, 10000

There are no consecutive perfect powers except (0,1) and (8,9), this is the proven Catalan’s conjecture (the additional properties of (8,9): the numbers *n* ≤ 10 with terminate reciprocal are {1, 2, 3, 4, 6, 8, 9, 10}, and the divisors of 10 are {1, 2, 3, 4, 6, 10}, the difference of these two sets are 8 and 9 (of course, the second set is a subset of the first set), (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these *n* except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2), besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest Ziesel number and the smallest integer such that the factorization of

over **Q** includes coefficients other than

(i.e. the 89th cyclotomic polynomial,

, is the first with coefficients other than

), besides, the repunit with length *k* (*R*_{k}) (where *k* = the concatenation of *n* and the unit (1), i.e. *k* = 10*n*+1) is prime for both *n* = 8 and *n* = 9, and not for any other *n* ≤ 1000, besides, the squares of 8 and 9 are the only two 2-digit automorphic numbers, besides, if we add zero (0, the additive identity) between 8 and 9, we get 809, the number ≤1000 whose Collatz sequence is longest, besides, 8 and 9 are the only two natural numbers *n* such that centered *n*-gonal numbers (the *k*th centered *n*-gonal number is *n*×*T*_{k}+1, where *T*_{k} is the *k*th triangular number) cannot be primes (8 is because all centered 8-gonal numbers are square numbers (4-gonal numbers), 9 is because all centered 9-gonal numbers are triangular numbers (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, (note that 8+4 (centered 8-gonal numbers and non-centered 4-gonal numbers) and 9+3 (centered 9-gonal numbers and non-centered 3-gonal numbers) are both 10), in fact, all polygonal numbers with rank > 2 are not primes, i.e. all primes *p* cannot be a polygonal number (except the trivial case, i.e. each *p* is the second *p*-gonal number)), assuming the Bunyakovsky conjecture is true. (i.e. 8 and 9 are the only two natural number *n* such that

is not irreducible) (Note that for *n* = 10, the centered 10-gonal numbers are exactly the star numbers)).