The smallest squares containingk 87's :
15876 = 1262, 1887876 = 13742, 87871876 = 93742,
878791878721 = 9374392, 3874187687878729 = 622429732.
872 = 7569, a square with different digits.
872 = 7569, 7 + 5 + 69 = 92,
872 = 7569, 7 + 569 = 242,
872 = 7569, 75 + 69 = 122.
872 = 93 + 143.
A cubic Polynomial : (X + 872)(X + 1442)(X + 3642) = X3 + 4012X2 + 625082X + 45601922.
Komachi equations:
872 = 98 * 76 + 5 * 4 * 3 * 2 + 1 = 9 + 87 * 65 * 4 / 3 + 2 * 10
= 9 + 8 * 7 / 6 * 54 * 3 / 2 * 10,
872 = - 12 - 22 * 32 - 42 * 52 + 62 + 72 + 892 = - 12 - 22 * 32 * 42 * 52 / 62 + 72 + 892
= - 12 + 22 * 32 - 42 * 52 - 62 + 72 + 892 = - 122 - 32 * 42 - 562 / 72 + 892
= 92 + 872 - 62 - 52 + 42 - 32 * 22 */ 12 = 92 + 872 - 62 - 52 - 42 - 32 + 22 + 12
= 92 + 872 + 62 + 542 / 32 - 212 = - 92 + 872 + 62 + 52 - 42 + 32 * 22 */ 12
= - 92 + 872 + 62 + 52 + 42 + 32 - 22 - 12 = - 92 + 872 - 62 - 542 / 32 + 212
= 92 * 82 * 72 / 62 + 52 * 42 + 32 + 22 + 102 = 982 / 72 * 62 + 52 * 42 + 32 + 22 + 102.
(872 - 2) = (32 - 2)(52 - 2)(72 - 2),
(872 - 3) = (92 - 3)(102 - 3) = (32 - 3)(42 - 3)(102 - 3),
(872 - 9) = (42 - 9)(62 - 9)(72 - 9) = (42 - 9)(332 - 9) = (62 - 9)(172 - 9).
872 + 882 + 892 + ... + 1362 = 7952.
12 + 22 + 32 + 42 + ... + 872 = 223300, which consists of digits 0, 2 and 3.
(1 + 2 + ... + 32)(33 + 34 + ... + 87) = 13202.
(13 + 23 + 33 + 43 + 53)(63 + 73 + ... + 123)(133 + 143 + ... + 873) = 43942502.
(12 + 22)(32)(42 + 52)(62 + ... + 872) = 202952,
(12 + ... + 62)(72 + ... + 382)(392 + ... + 632)(642 + ... + 872) = 1255399602,
(12 + ... + 132)(142 + ... + 582)(592)(602 + ... + 872) = 1696067102,
(12 + ... + 222)(232 + ... + 492)(502 + ... + 592)(602 + ... + 872) = 7961530502.
877 = 37725479487783, 37 + 7254 + 7 + 94 + 87 + 7 + 83 = 872,
878 = 3282116715437121,
3+2+8+2+1+1+6+7154+371+21 = 3+2+8+21+1+6+7154+371+2+1
= 3+2+8+211+67+154+3+7121 = 3+2+82+11+6715+43+712+1
= 3+2+821+1+6715+4+3+7+12+1 = 3+28+2+1+1+6+7154+371+2+1
= 3+282+11+6715+437+121 = 3+282+116+7154+3+7+1+2+1
= 32+8+211+6+7154+37+121 = 328+2+1+1+6+7154+3+71+2+1
= 328+2+1+1+67+1+5+43+7121 = 328+21+1+6+7154+37+1+21
= 328+21+16+71+5+4+3+7121 = 328+21+16+7154+37+12+1 = 872,
879 = 285544154243029527,
2+8+5+5+4+4154+2+430+2952+7 = 2+8+5+5+4+4154+2430+2+952+7 = 872,
= 2+8+5+5+4415+42+4+3029+52+7 = 2+8+55+4415+4+24+3029+5+27
= 2+8+55+4415+42+4+3029+5+2+7 = 2+8+5544+1542+430+2+9+5+27
= 2+8+5544+1542+430+29+5+2+7 = 2+855+4+4154+2430+2+95+27
= 28+5+5+4415+4+24+3029+52+7 = 28+55+4415+4+2+4+3029+5+27
= 28+55+4415+4+24+3029+5+2+7 = 28+5544+1542+430+2+9+5+2+7
= 2855+4+4154+2+430+2+95+27 = 2855+441+5+4243+02+9+5+2+7 = 872.
872 = 7569 appears in the decimal expressions of π and e:
π = 3.14159•••7569••• (from the 18176th digit),
e = 2.71828•••7569••• (from the 9118th digit).
Page of Squares : First Upload March 8, 2004 ; Last Revised April 6, 2010
by Yoshio Mimura, Kobe, Japan