## The Two Special Sets of Numbers

Posted: May 3, 2017 in Mathematics

Sum of Zeroth Powers

$1^0 + 6^0 + 7^0 + 23^0 + 24^0 + 30^0 + 38^0 + 47^0 + 54^0 + 55^0 = 10$

$2^0 + 3^0 + 10^0 + 19^0 + 27^0 + 33^0 + 34^0 + 50^0 + 51^0 + 56^0 = 10$

Sum of First Powers

$1^1 + 6^1 + 7^1 + 23^1 + 24^1 + 30^1 + 38^1 + 47^1 + 54^1 + 55^1 = 285$

$2^1 + 3^1 + 10^1 + 19^1 + 27^1 + 33^1 + 34^1 + 50^1 + 51^1 + 56^1 = 285$

Sum of Second Powers

$1^2 + 6^2 + 7^2 + 23^2 + 24^2 + 30^2 + 38^2 + 47^2 + 54^2 + 55^2 = 11685$

$2^2 + 3^2 + 10^2 + 19^2 + 27^2 + 33^2 + 34^2 + 50^2 + 51^2 + 56^2 = 11685$

Sum of Third Powers

$1^3 + 6^3 + 7^3 + 23^3 + 24^3 + 30^3 + 38^3 + 47^3 + 54^3 + 55^3 = 536085$

$2^3 + 3^3 + 10^3 + 19^3 + 27^3 + 33^3 + 34^3 + 50^3 + 51^3 + 56^3 = 536085$

Sum of Fourth Powers

$1^4 + 6^4 + 7^4 + 23^4 + 24^4 + 30^4 + 38^4 + 47^4 + 54^4 + 55^4 = 26043813$

$2^4 + 3^4 + 10^4 + 19^4 + 27^4 + 33^4 + 34^4 + 50^4 + 51^4 + 56^4 = 26043813$

Sum of Fifth Powers

$1^5 + 6^5 + 7^5 + 23^5 + 24^5 + 30^5 + 38^5 + 47^5 + 54^5 + 55^5 = 1309753125$

$2^5 + 3^5 + 10^5 + 19^5 + 27^5 + 33^5 + 34^5 + 50^5 + 51^5 + 56^5 = 1309753125$

Sum of Sixth Powers

$1^6 + 6^6 + 7^6 + 23^6 + 24^6 + 30^6 + 38^6 + 47^6 + 54^6 + 55^6 = 67334006805$

$2^6 + 3^6 + 10^6 + 19^6 + 27^6 + 33^6 + 34^6 + 50^6 + 51^6 + 56^6 = 67334006805$

Sum of Seventh Powers

$1^7 + 6^7 + 7^7 + 23^7 + 24^7 + 30^7 + 38^7 + 47^7 + 54^7 + 55^7 = 3512261547765$

$2^7 + 3^7 + 10^7 + 19^7 + 27^7 + 33^7 + 34^7 + 50^7 + 51^7 + 56^7 = 3512261547765$

Sum of Eighth Powers

$1^8 + 6^8 + 7^8 + 23^8 + 24^8 + 30^8 + 38^8 + 47^8 + 54^8 + 55^8 = 185039471773893$

$2^8 + 3^8 + 10^8 + 19^8 + 27^8 + 33^8 + 34^8 + 50^8 + 51^8 + 56^8 = 185039471773893$