## Friday, October 17, 2014

### Interesting Integers - Armstrong Numbers

Armstrong numbers are n digit integers that are are equal to the sum of the nth power of their digits.  For example 370 has 3 digits and 370 = 33 + 73 + 03 = 27 + 343 + 0.

Other examples of Armstrong numbers are:
1,634 = 14 +64 + 34 + 44
54,748 = 55 + 45 + 75 + 45 + 85
548,834 = 56 +46 + 86 + 86 + 36 + 46
1,741,725 = 17 + 77 + 47 + 17 + 77 + 27 + 57
24,678,050 = 28 + 48 + 68 + 78 + 88 + 08 + 58 + 08
534,494,836 = 59 + 39 + 49 + 49 + 99 + 49 + 89 + 39 + 69
4,679,307,774 = 410 + 610 + 710 + 910 + 310 + 010 + 710 + 710 + 710 + 410
94,204,591,914 = 911 + 411 + 211 + 011 + 411 + 511 + 911 + 111 + 911 + 111 + 411
28,116,440,335,967 = 214 + 814 + 114 + 114 + 614 + 414 + 414 + 014 + 314 + 314 + 514 + 914 + 614 + 714
4,338,281,769,391,370 = 416 + 316 + 316 + 816 + 216 + 816 + 116 + 716 + 616 + 916 + 316 + 916 + 116 + 316 + 716 + 016
35,875,699,062,250,035 = 317 + 517 + 817 + 717 + 517 + 617 + 917 + 917 + 017 + 617 + 217 + 217 + 517 + 017 + 017 + 317 + 517
1,517,841,543,307,505,039 = 119 + 519 + 119 + 719 + 819 + 419 + 119 + 519 + 419 + 319 + 319 + 019 + 719 + 519 + 019 + 519 + 019 + 319 + 919
63,105,425,988,599,693,916 = 620 + 320 + 120 + 020 + 520 + 420 + 220 + 520 + 920 + 820 + 820 + 520 + 920 + 920 + 620 + 920 + 320 + 920 + 20 + 1620
128,468,643,043,731,391,252 = 121 + 221 + 821 + 421 + 621 + 821 + 621 + 421 + 321 + 021 + 421 + 321 + 721 + 321 + 121 + 321 + 921 + 121 + 221 + 521 + 221
35,452,590,104,031,691,935,943 = 323 + 523 + 423 + 523 + 223 + 523 + 923 + 023 + 123 + 023 + 423 + 023 + 323 + 123 + 623 + 923 + 123 + 923 + 323 + 523 + 923 + 423 + 323
174,088,005,938,065,293,023,722 = 124 + 724 + 424 + 024 + 824 + 824 + 024 + 024 + 524 + 924 + 324 + 824 + 024 + 624 + 524 + 224 + 924 + 324 + 024 + 224 + 324 + 724 + 224 + 224
4,422,095,118,095,899,619,457,938 = 425 + 425 + 225 + 225 + 025 + 925 + 525 + 125 + 125 + 825 + 025 + 925 + 525 + 825 + 925 + 925 + 625 + 125 + 925 + 425 + 525 + 725 + 925 + 325 + 825
121,204,998,563,613,372,405,438,066 = 127 + 227 + 127 + 227 + 027 + 427 + 927 + 927 + 827 + 527 + 627 + 327 + 627 + 127 + 327 + 327 + 727 + 227 + 427 + 027 + 527 + 427 + 327 + 827 + 027 + 627 + 627
23,866,716,435,523,975,980,390,369,295 = 229 + 329 + 829 + 629 + 629 + 729 + 129 + 629 + 429 + 329 + 529 + 529 + 229 + 329 + 929 + 729 + 529 + 929 + 829 + 029 + 329 + 929 + 029 + 329 + 629 + 929 + 229 + 929 + 529
1,145,037,275,765,491,025,924,292,050,346 = 131 + 131 + 431 + 531 + 031 + 331 + 731 + 231 + 731 + 531 + 731 + 631 + 531 + 431 + 931 + 131 + 031 + 231 + 531 + 931 + 231 + 431 + 231 + 931 + 231 + 031 + 531 + 031 + 331 + 431 + 631
17,333,509,997,782,249,308,725,103,962,772 = 132 + 732 + 332 + 332 + 332 + 532 + 032 + 932 + 932 + 932 + 732 + 732 + 832 + 232 + 232 + 432 + 932 + 332 + 032 + 832 + 732 + 232 + 532 + 132 + 032 + 332 + 932 + 632 + 232 + 732 + 732 + 232
186,709,961,001,538,790,100,634,132,976,990 = 133 + 833 + 633 + 733 + 033 + 933 + 933 + 633 + 133 + 033 + 033 + 133 + 533 + 333 + 833 + 733 + 933 + 033 + 133 + 033 + 033 + 633 + 333 + 433 + 133 + 333 + 233 + 933 + 733 + 633 + 933 + 933 + 033
1,122,763,285,329,372,541,592,822,900,204,593 = 134 + 134 + 234 + 234 + 734 + 634 + 334 + 234 + 834 + 534 + 334 + 234 + 934 + 334 + 734 + 234 + 534 + 434 + 134 + 534 + 934 + 234 + 834 + 234 + 234 + 934 + 034 + 034 + 234 + 034 + 434 + 534 + 934 + 334
12,639,369,517,103,790,328,947,807,201,478,392 = 135 + 235 + 635 + 335 + 935 + 335 + 635 + 935 + 535 + 135 + 735 + 135 + 035 + 335 + 735 + 935 + 035 + 335 + 235 + 835 + 935 + 435 + 735 + 835 + 035 + 735 + 235 + 035 + 135 + 435 + 735 + 835 + 335 + 935 + 235
1,219,167,219,625,434,121,569,735,803,609,966,019 = 137 + 237 + 137 + 937 + 137 + 637 + 737 + 237 + 137 + 937 + 637 + 237 + 537 + 437 + 337 + 437 + 137 + 237 + 137 + 537 + 637 + 937 + 737 + 337 + 537 + 837 + 037 + 337 + 637 + 037 + 937 + 937 + 637 + 637 + 037 + 137 + 937
12,815,792,078,366,059,955,099,770,545,296,129,367 = 138 + 238 + 838 + 138 + 538 + 738 + 938 + 238 + 038 + 738 + 838 + 338 + 638 + 638 + 038 + 538 + 938 + 938 + 538 + 538 + 038 + 938 + 938 + 738 + 738 + 038 + 538 + 438 + 538 + 238 + 938 + 638 + 138 + 238 + 938 + 338 + 638 + 738
115,132,219,018,763,992,565,095,597,973,971,522,401 = 139 + 139 + 539 + 139 + 339 + 239 + 239 + 139 + 939 + 039 + 139 + 839 + 739 + 639 + 339 + 939 + 939 + 239 + 539 + 639 + 539 + 039 + 939 + 539 + 539 + 939 + 739 + 939 + 739 + 339 + 939 + 739 + 139 + 539 + 239 + 239 + 439 + 039 + 139
The number above is the largest Armstrong number.  It has been proven that there are only 88 Armstrong numbers.

The number below is larger, but it is not an Armstrong number.  Can you tell me why?  I have included it here because it is still an interesting integer - even if it is not an Armstrong number.

36,428,594,490,313,158,783,584,452,532,870,892,261,556 = 342 + 642 + 442 + 242 + 842 + 542 + 942 + 442 + 442 + 942 + 042 + 342 + 142 + 342 + 142 + 542 + 842 + 742 + 842 + 342 + 542 + 842 + 442 + 442 + 542 + 242 + 542 + 342 + 242 + 842 + 742 + 042 + 842 + 942 + 242 + 242 + 642 + 142 + 542 + 542 + 642

REFERENCES:

David