The smallest squares containingk 97's :
30976 = 1762, 7974976 = 28242, 19789736976 = 1406762,
19769797112976 = 44463242, 6979729763979729 = 835447772.
97 is the sum of m squares for m = 2,3,4,..,83.
97 is the fifth prime for which the Legendre Symbol (a/97) = 1 for a = 1, 2, 3, 4.
972 is the second square which is the sum of 6 fifth powers : 15 + 15 + 25 + 55 + 55 + 55.
972 = 44 + 64 + 64 + 94.
978 = 7837433594376961, 7 + 8 + 3 + 743 + 3 + 5 + 943 + 7696 + 1 = 972,
7 + 8374 + 3 + 3 + 5 + 9 + 4 + 37 + 6 + 961 =7 + 8374 + 3 + 35 + 9 + 4 + 3 + 7 + 6 + 961
= 7 + 8374 + 33 + 5 + 9 + 4 + 3 + 7 + 6 + 961 = 783 + 7433 + 59 + 437 + 696 + 1
= 7837 + 433 + 5 + 94 + 3 + 76 + 961 = 7837 + 433 + 59 + 43 + 76 + 961 = 972.
13 + 333 + 653 + 973 = 11062.
Komachi Fraction : 972 = 2879154/306.
Komachi equations:
972 = 1 * 2 * 3 * 4 * 56 * 7 - 8 + 9 = - 9 + 8 + 76 * 5 + 43 * 210,
972 = - 12 / 22 * 342 + 52 + 672 + 82 * 92.
(972 + 7) = (92 + 7)(102 + 7) = (12 + 7)(22 + 7)(102 + 7).
12 + 22 + 32 + 42 + ... + 972 = 308945, which consists of different digits.
(1 + 2 + ... + 9)(10 + 11 + ... + 45)(46 + 47 + ... + 97) = 128702,
(1 + 2 + ... + 24)(25 + 26 + ... + 48)(49 + 50 + ... + 97) = 306602,
(1 + 2 + ... + 36)(37)(38 + 39 + ... + 97) = 99902,
(1 + 2 + ... + 44)(45)(46 + 47 + ... + 97) = 128702.
972 = 9409 appears in the decimal expressions of π and e:
π = 3.14159•••9409••• (from the 2983rd digit),
e = 2.71828•••9409••• (from the 5510th digit).
Page of Squares : First Upload March 22, 2004 ; Last Revised April 9, 2010
by Yoshio Mimura, Kobe, Japan