Difference of Powers

This page will outline the interesting world of differences of powers.

What is "difference of powers"?
Difference of powers is a term I came up with to describe the result of the expression xn - yn, where n is a whole number and x and y are two consecutive whole numbers. The interesting thing about difference of powers is that their pattern of increasing always involves a constant.

Powers
This is the list of powers for every value of n (up to 4 so far). On the right hand side of each equation in the list, the first number is the power itself, the second number (the one in parentheses) is the difference from the previous power, the third is the difference of difference, the fourth number is the increase in the difference of differences, and the fifth number (the one in brackets) is the amount it increased compared to the last increase.

2

 * 1^2 = 1; (1); 1; +1; [1]
 * 2^2 = 4; (3); 2; +1; [0]
 * 3^2 = 9; (5); 2; +0; [-1]
 * 4^2 = 16; (7); 2; +0; [0]
 * 5^2 = 25; (9); 2; +0; [0]
 * 6^2 = 36; (11); 2; +0; [0]
 * 7^2 = 49; (13); 2; +0; [0]
 * 8^2 = 64; (15); 2; +0; [0]
 * 9^2 = 81; (17); 2; +0; [0]
 * 10^2 = 100; (19); 2; +0; [0]

3

 * 1^3 = 1; (1); 1; +1; [1]
 * 2^3 = 8; (7); 6; +5; [4]
 * 3^3 = 27; (19); 12; +6; [1]
 * 4^3 = 64; (37); 18; +6; [0]
 * 5^3 = 125; (61); 24; +6; [0]
 * 6^3 = 216; (91); 30; +6; [0]
 * 7^3 = 343; (127); 36; +6; [0]
 * 8^3 = 512; (169); 42; +6; [0]
 * 9^3 = 729; (217); 48; +6; [0]
 * 10^3 = 1,000; (271); 54; +6; [0]

4

 * 1^4 = 1; (1); 1; +1; [1]
 * 2^4 = 16; (15); 14; +13; [12]
 * 3^4 = 81; (65); 50; +36; [23]
 * 4^4 = 256; (175); 110; +60; [24]
 * 5^4 = 625; (369); 194; +84; [24]
 * 6^4 = 1,296; (671); 302; +108; [24]
 * 7^4 = 2,401; (1,105); 434; +132; [24]
 * 8^4 = 4,096; (1,695); 590; +156; [24]
 * 9^4 = 6,561; (2,465); 770; +180; [24]
 * 10^4 = 10,000; (3,439); 974; +204; [24]