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 will always result in the involvement of a constant, which appears to be n! (the factorial of n).

Powers
This is the list of powers for every value of n (up to 5 so far). The five numbers in parentheses are related to the comparisons of each power; the first is the difference of the current power and the previous power, the second is the difference of the current power and previous power and the difference of the previous power and the power before that, the third is the increase in the difference of differences, the fourth is the amount it increased compared to the last increase, and the fifth is the difference in amount increased.

2

 * 12 = 1 (1, 1, +1, 1, 1)
 * 22 = 4 (3, 2, +1, 0, -1)
 * 32 = 9 (5, 2, +0, -1, -1)
 * 42 = 16 (7, 2, +0, 0, 1)
 * 52 = 25 (9, 2, +0, 0, 0)
 * 62 = 36 (11, 2, +0, 0, 0)
 * 72 = 49 (13, 2, +0, 0, 0)
 * 82 = 64 (15, 2, +0, 0, 0)
 * 92 = 81 (17, 2, +0, 0, 0)
 * 102 = 100 (19, 2, +0, 0, 0)

3

 * 13 = 1 (1, 1, +1, 1, 1)
 * 23 = 8 (7, 6, +5, 4, 3)
 * 33 = 27 (19, 12, +6, 1, -2)
 * 43 = 64 (37, 18, +6, 0, 2)
 * 53 = 125 (61, 24, +6, 0, 0)
 * 63 = 216 (91, 30, +6, 0, 0)
 * 73 = 343 (127, 36, +6, 0, 0)
 * 83 = 512 (169, 42, +6, 0, 0)
 * 93 = 729 (217, 48, +6, 0, 0)
 * 103 = 1,000 (271, 54, +6, 0, 0)

4

 * 14 = 1 (1, 1, +1, 1, 1)
 * 24 = 16 (15, 14, +13, 12, 11)
 * 34 = 81 (65, 50, +36, 23, 12)
 * 44 = 256 (175, 110, +60, 24, 1)
 * 54 = 625 (369, 194, +84, 24, 0)
 * 64 = 1,296 (671, 302, +108, 24, 0)
 * 74 = 2,401 (1,105, 434, +132, 24, 0)
 * 84 = 4,096 (1,695, 590, +156, 24, 0)
 * 94 = 6,561 (2,465, 770, +180, 24, 0)
 * 104 = 10,000 (3,439, 974, +204, 24, 0)

5

 * 15 = 1 (1, 1, +1, 1, 1)
 * 25 = 32 (31, 30, +29, 28, 27)
 * 35 = 243 (211, 180, +150, 121, 93)
 * 45 = 1,024 (781, 570, +360, 210, 89)
 * 55 = 3,125 (2,101, 1,320, +750, 390, 180)
 * 65 = 7,776 (4,651, 2,550, +1,230, 480, 90)
 * 75 = 16,807 (9,031, 4,380, +1,830, 600, 120)
 * 85 = 32,768 (15,961, 6,930, +2,550, 720, 120)
 * 95 = 59,049 (26,281, 10,320, +3,390, 840, 120)
 * 105 = 100,000 (40,951, 14,670 +4,350, 960, 120)