Pluperfect Digital Invariant/Examples/39 Digits

Examples of $39$-Digit Pluperfect Digital Invariants

The $39$-digit pluperfect digital invariants are:

\(\ds \)

\(\)

\(\ds 115 \, 132 \, 219 \, 018 \, 763 \, 992 \, 565 \, 095 \, 597 \, 973 \, 971 \, 522 \, 400\)

\(\ds \)

\(=\)

\(\ds 1 + 1 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 + 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 549 \, 755 \, 813 \, 888 + 1 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 0 + 1 + 166 \, 153 \, 499 \, 473 \, 114 \, 484 \, 112 \, 975 \, 882 \, 535 \, 043 \, 072\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 2 \, 227 \, 915 \, 756 \, 473 \, 955 \, 677 \, 973 \, 140 \, 996 \, 096\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 2 \, 227 \, 915 \, 756 \, 473 \, 955 \, 677 \, 973 \, 140 \, 996 \, 096\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 0 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143 + 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289 + 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 549 \, 755 \, 813 \, 888 + 302 \, 231 \, 454 \, 903 \, 657 \, 293 \, 676 \, 544 + 0 + 0\)

\(\ds \)

\(=\)

\(\ds 1^{39} + 1^{39} + 5^{39} + 1^{39} + 3^{39} + 2^{39} + 2^{39} + 1^{39} + 9^{39} + 0^{39} + 1^{39} + 8^{39} + 7^{39}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 6^{39} + 3^{39} + 9^{39} + 9^{39} + 2^{39} + 5^{39} + 6^{39} + 5^{39} + 0^{39} + 9^{39} + 5^{39} + 5^{39} + 9^{39}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 7^{39} + 9^{39} + 7^{39} + 3^{39} + 9^{39} + 7^{39} + 1^{39} + 5^{39} + 2^{39} + 2^{39} + 4^{39} + 0^{39} + 0^{39}\)

\(\ds \)

\(\)

\(\ds 115 \, 132 \, 219 \, 018 \, 763 \, 992 \, 565 \, 095 \, 597 \, 973 \, 971 \, 522 \, 401\)

\(\ds \)

\(=\)

\(\ds 1 + 1 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 + 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 549 \, 755 \, 813 \, 888 + 1 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 0 + 1 + 166 \, 153 \, 499 \, 473 \, 114 \, 484 \, 112 \, 975 \, 882 \, 535 \, 043 \, 072\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 2 \, 227 \, 915 \, 756 \, 473 \, 955 \, 677 \, 973 \, 140 \, 996 \, 096\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 2 \, 227 \, 915 \, 756 \, 473 \, 955 \, 677 \, 973 \, 140 \, 996 \, 096\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 0 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143 + 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143 + 4 \, 052 \, 555 \, 153 \, 018 \, 976 \, 267\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 16 \, 423 \, 203 \, 268 \, 260 \, 658 \, 146 \, 231 \, 467 \, 800 \, 709 \, 255 \, 289 + 909 \, 543 \, 680 \, 129 \, 861 \, 140 \, 820 \, 205 \, 019 \, 889 \, 143\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 1 + 1 \, 818 \, 989 \, 403 \, 545 \, 856 \, 475 \, 830 \, 078 \, 125 + 549 \, 755 \, 813 \, 888\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 549 \, 755 \, 813 \, 888 + 302 \, 231 \, 454 \, 903 \, 657 \, 293 \, 676 \, 544 + 0 + 1\)

\(\ds \)

\(=\)

\(\ds 1^{39} + 1^{39} + 5^{39} + 1^{39} + 3^{39} + 2^{39} + 2^{39} + 1^{39} + 9^{39} + 0^{39} + 1^{39} + 8^{39} + 7^{39}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 6^{39} + 3^{39} + 9^{39} + 9^{39} + 2^{39} + 5^{39} + 6^{39} + 5^{39} + 0^{39} + 9^{39} + 5^{39} + 5^{39} + 9^{39}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds 7^{39} + 9^{39} + 7^{39} + 3^{39} + 9^{39} + 7^{39} + 1^{39} + 5^{39} + 2^{39} + 2^{39} + 4^{39} + 0^{39} + 1^{39}\)

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $115,132,219,018,763,992,565,095,597,973,971,522,401$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $115,132,219,018,763,992,565,095,597,973,971,522,401$

• Weisstein, Eric W. "Narcissistic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NarcissisticNumber.html