The smallest squares containingk 78's :
784 = 282, 278784 = 5282, 39789078784 = 1994722,
40781787867844 = 63860622, 8782787807807841 = 937165292.
782 = 6084, a zigzag square with different and even digits.
782± 5 are primes.
782 = (1)(2)(3 + 4 + ... + 6)(7 + 8 + ... + 19).
782 + 792 + 802 + 812 + 822 + 832 + 842 = 852 + 862 + 872 + 882 + 892 + 902.
782 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103 + 113 + 123.
782 = 6084, 60 + 84 = 122.
782 = 83 + 133 + 153,
782 is the 9th square which is the sum of 7 fourth powers : [2,2,3,5,5,5,8], [4,4,5,5,5,6,7].
78k + 1092k + 1950k + 10569k are squares for k = 1,2,3 (1172, 108032, 10905572).
Komachi equations:
782 = 1 - 2 + 3 - 4 * 5 + 678 * 9 = 1 - 2 - 3 * 4 - 5 + 678 * 9
= 1 - 2 * 3 * 4 + 5 + 678 * 9 = 1 + 2345 + 6 * 7 * 89
= 1 + 2 * 3 * 4 * 5 + 67 * 89 = - 12 + 3 - 4 - 5 + 678 * 9
= 9 + 8 * 765 - 43 - 2 */ 1 = - 9 * 8 + 76 * 54 * 3 / 2 */ 1
= - 9 + 8 * 765 + 4 - 32 + 1 = 9 * 8 * 7 + 65 * 43 * 2 - 10
= 987 * 6 - 5 - 43 + 210 = - 9 + 8 * 765 - 4 - 3 - 2 * 10,
782 = 12 / 22 * 342 * 52 * 62 - 72 * 82 * 92 = 12 + 22 - 32 - 42 - 52 - 62 + 782 + 92
= 12 * 22 * 32 - 42 + 52 + 62 + 782 - 92 = - 12 - 22 + 32 + 42 + 52 + 62 + 782 - 92
= - 12 * 22 * 32 + 42 - 52 - 62 + 782 + 92 = 982 + 72 + 62 - 52 * 42 * 32 - 22 - 12
= 92 + 82 * 72 - 62 + 542 - 32 - 22 */ 12.
(782 + 6) = (32 + 6)(202 + 6) = (82 + 6)(92 + 6) = (12 + 6)(22 + 6)(92 + 6),
(782 - 8) = (62 - 8)(152 - 8).
(1 + 2 + ... + 20)(21 + 22 + ... + 56)(57 + 58 + ... + 78) = 207902,
(1 + 2 + ... + 21)(22 + 23 + ... + 33)(34 + 35 + ... + 78) = 138602,
(1 + 2 + ... + 27)(28 + 29 + ... + 42)(43 + 44 + ... + 78) = 207902,
(1 + 2 + ... + 27)(28 + 29 + ... + 71)(72 + 73 + ... + 78) = 207902,
(1 + 2 + ... + 35)(36 + 37 + ... + 50)(51 + 52 + ... + 78) = 270902.
(13 + 23 + ... + 53)(63 + 73 + ... + 503)(513 + 523 + ... + 783) = 536382002.
786 = 225199600704, 2 + 2 + 51 + 9 + 9 + 6007 + 04 = 782,
787 = 17565568854912, 1 + 7 + 5 + 6 + 556 + 8 + 8 + 5491 + 2 = 782,
788 = 1370114370683136, 1370 + 11 + 4370 + 6 + 8 + 313 + 6 = 782,
789 = 106868920913284608,
1+068+68+920+91+328+4608 = 10+6+8+6+89+20+9+1328+4608
= 10+6+8+68+9+2091+3284+608 = 10+6+86+8+9+20+9+1328+4608
= 10+68+6+8+9+2091+3284+608 = 106+8+6+8+9+2+09+1328+4608 = 782.
782 = 6084 appears in the decimal expressions of π and e:
π = 3.14159•••6084••• (from the 1860th digit),
e = 2.71828•••6084••• (from the 4054th digit).
Page of Squares : First Upload February 23, 2004 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan