The smallest squares containingk 80's :
2809 = 532, 5280804 = 22982, 1568080801 = 395992,
7180802808804 = 26797022, 3780180808080625 = 614831752.
Komachi equations:
802 = - 98 / 7 - 6 + 5 * 4 * 321 = 9 - 8 - 7 + 6 + 5 * 4 * 32 * 10
= - 9 + 8 + 7 - 6 + 5 * 4 * 32 * 10,
802 = 122 * 32 - 42 - 562 / 72 + 82 * 92 = - 12 * 22 + 342 + 562 / 72 + 82 * 92
= - 982 / 72 + 62 + 542 * 32 / 22 - 12 = - 92 - 82 + 72 - 62 * 52 + 432 * 22 */ 12
= 92 * 82 - 72 * 62 + 542 - 32 * 22 + 102 = - 982 / 72 - 62 * 52 + 432 * 22 + 102,
802 = - 14 + 24 + 34 - 44 - 564 / 74 / 84 + 94.
80 and 81 are consecutive integers having square factors (the 7th case).
802 = 43 + 83 + 123 + 163.
802 = (22 + 4)(42 + 4)(62 + 4).
(802 + 8) = (82 + 8)(92 + 8), (802 + 9) = (22 + 9)(222 + 9).
4132 = 222 + 232 + 242 + ... + 802.
(1 + 2 + 3 + 4)(5 + 6 + ... + 55)(56 + 57 + ... + 80) = 51002,
(1 + 2 + ... + 17)(18 + 19 + ... + 38)(39 + 40 + ... + 80) = 149942,
(1 + 2 + ... + 24)(25 + 26 + ... + 45)(46 + 47 + ... + 80) = 220502.
802 = 6400 appears in the decimal expressions of π and e:
π = 3.14159•••6400••• (from the 5091st digit),
e = 2.71828•••6400••• (from the 6357th digit).
Page of Squares : First Upload March 1, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan