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Stifled Squares And (nth roots)

written by Adi Cox

9th March 2015

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Squares

x x | x.(2x)^1 + x.(2x)^0 = 2x^2 + x_____|_________________________________________________________1 1 | [1].(2[1])^1 + [1].(2[1])^0 = 2 + 1 = 3 = 11 base 022 2 | [2].(2[2])^1 + [2].(2[2])^0 = 8 + 2 = 10 = 22 base 043 3 | [3].(2[3])^1 + [3].(2[3])^0 = 18 + 3 = 21 = 33 base 064 4 | [4].(2[4])^1 + [4].(2[3])^0 = 32 + 4 = 36 = 44 base 085 5 | [5].(2[5])^1 + [5].(2[3])^0 = 50 + 5 = 55 = 55 base 106 6 | [6].(2[6])^1 + [6].(2[3])^0 = 72 + 6 = 78 = 66 base 12

7 7 | [7].(2[7])^1 + [7].(2[3])^0 = 98 + 7 = 105 = 77 base 148 8 | [8].(2[8])^1 + [8].(2[3])^0 = 128 + 8 = 136 = 88 base 169 9 | [9].(2[9])^1 + [9].(2[3])^0 = 169 + 9 = 171 = 99 base 18. .. .. .

nth roots (where x is the nth root of r(n,x))

E.g. 162723 = 33333 base 15

3.15^4 + 3.15^3 + 3.15^2 + 3.15^1 + 3.15^0

151875 + 010125 + 000675 + 000045 + 000003 = 162723

r(n,x) = n^(n-1).x^(n-0) + n^(n-2).x^(n-1) + ... n^0.x^1

| n 2 3 4 5 ..._____|______________________________________________________________ x |

| 1 | 3 13 85 781 2 | 10 86 1170 22222 3 | 21 273 5655 162723

4 | 36 628 17476 673684 5 | 55 1205 42105 2034505 6 | 78 2058 86550 5027586 7 | 105 3241 159355 10813327 8 | 136 4808 270600 21005128 9 | 171 6813 431901 37744389 10 | 210 9310 656410 63775510 11 | 253 12353 958815 102520891 12 | 300 15996 1355340 158155932 13 | 351 20293 1863745 235684033 14 | 406 25298 2503326 341011594 15 | 465 31065 3294915 481023015 16 | 528 37648 4260880 663655696

17 | 595 45101 5425125 897975037 18 | 666 53478 6813090 1194249438 19 | 741 62833 8451751 1564025299

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20 | 820 73220 10369620 2020202020

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