360
The smallest squares containing k 360's :
3600 = 602,
36036009 = 60032,
36036081036036 = 60030062.
3602 = (22 - 1)(112 - 1)(192 - 1) = (22 - 1)(32 - 1)(42 - 1)(192 - 1) = (22 - 1)(42 - 1)(52 - 1)(112 - 1)
= (32 - 1)(52 - 1)(262 - 1) = (42 - 1)(52 - 1)(192 - 1).
Komachi equations:
3602 = 92 + 82 - 72 + 62 * 52 * 42 * 32 + 22 - 102 = - 92 - 82 + 72 + 62 * 52 * 42 * 32 - 22 + 102.
(1 + 2 + 3 + 4)(5 + 6 + 7)(8)(9)(10) = 3602,
(1 + 2 + 3)(4 + 5 + 6)(7 + 8 + 9)(10 + 11 + 12 + 13 + 14) = 3602,
(1)(2)(3)(4 + 5 + 6)(7 + 8 + 9)(10 + 11 + 12 + 13 + 14) = 3602,
(1 + 2 + 3 + 4 + 5)(6)(7 + 8 + 9)(10 + 11 + 12 + 13 + 14) = 3602,
(1 + 2 + 3 + ... + 9)(10 + 11 + 12 + ... + 14)(15 + 16 + 17) = 3602,
(1 + 2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + ... + 27) = 3602,
(1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 69) = 3602.
by Yoshio Mimura, Kobe, Japan
361
the square of 19.
The smallest squares containing k 361's :
361 = 192,
361722361 = 190192,
10036103616361 = 31679812.
The squares which begin with 361 and end in 361 are
361722361 = 190192, 36107220361 = 1900192, 36187833361 = 1902312,
361178162361 = 6009812, 361223838361 = 6010192,...
3612 is the first square which is the sum of 10 sixth powers.
3612 = 130321, 13 + 33 + 03 + 33 + 23 + 13 = 82,
3612 = 130321, 1 + 30 + 32 + 1 = 82,
3615 = 6131066257801, 62 + 12 + 32 + 102 + 62 + 62 + 22 + 52 + 72 + 82 + 02 + 12 = 361.
3-by-3 magic squares consisting of different squares with constant 3612:
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361 = 192, 36 = 62, 1 = 12.
1782 + 1792 + 1802 + 1812 + 1822 + ... + 3612 = 37262.
Page of Squares : First Upload January 7, 2005 ; Last Revised February 16, 2009by Yoshio Mimura, Kobe, Japan
362
The smallest squares containing k 362's :
362404 = 6022,
36268536249 = 1904432,
3628362323620321 = 602358892.
254k + 362k + 394k + 590k are squares for k = 1,2,3 (402, 8362, 181762).
10k + 164k + 362k + 833k are squares for k = 1,2,3 (372, 9232, 250972).
Komachi Fraction : 3622 = 9435168 / 72.
Komachi equations:
3622 = 9 * 8 * 7 * 65 * 4 + 3 + 2 - 1 = 9 * 8 * 7 * 65 * 4 - 3 * 2 + 10,
3622 = 12 * 22 + 32 * 452 + 62 * 72 * 82 - 92.
The third square which is the sum of a prime and a square in 8 ways :
32 + 353, 52 + 337, 72 + 313, 92 + 281, 112 + 241, 132 + 193, 152 + 137, 172 + 73.
3622 = 131044, 1 + 3 + 1 + 0 + 44 = 72,
3622 = 131044, 13 + 104 + 4 = 112.
(13 + 23 + 33 + ... + 243)(253 + 263 + 273 + ... + 873)(883 + 893 + 903 + ... + 3623) = 750934800002.
3622 = 131044 appears in the decimal expression of e:
e = 2.71828•••131044••• (from the 27867th digit).
by Yoshio Mimura, Kobe, Japan
363
The smallest squares containing k 363's :
93636 = 3062,
3636813636 = 603062,
174363943633636 = 132046942.
3632 = 131769, a zigzag square.
Cubic Polynomial : (X + 3632)(X + 13642)(X + 20162) = X3 + 24612X2 + 28882922X + 9981861122.
3632 = 131769, 1 + 3 + 1 + 7 + 69 = 92,
3632 = 131769, 1 + 3 + 17 + 6 + 9 = 62,
3632 = 131769, 13 + 1 + 7 + 6 + 9 = 62,
3632 = 131769, 131 + 769 = 302.
3632 = 223 + 333 + 443.
(13 + 23 + 33 + ... + 1113)(1123 + 1133 + 1143 + ... + 1683)(1693 + 1703 + 1713 + ... + 3633) = 51188021539202.
3-by-3 magic squares consisting of different squares with constant 3632:
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3632 = 131769 appears in the decimal expressions of π and e:
π = 3.14159•••131769••• (from the 86894th digit),
e = 2.71828•••131769••• (from the 102122nd digit)
by Yoshio Mimura, Kobe, Japan
364
The smallest squares containing k 364's :
3364 = 582,
503643364 = 224422,
636436443646864 = 252276922.
The squares which begin with 364 and end in 364 are
36458847364 = 1909422, 364142247364 = 6034422, 364282259364 = 6035582,
364745939364 = 6039422, 364886067364 = 6040582,...
3642 = 132496, a square with different digits.
3642 = (22 + 3)(72 + 3)(192 + 3).
Cubic Polynomial : (X + 872)(X + 1442)(X + 3642) = X3 + 4012X2 + 625082X + 45601922.
10101k + 12103k + 28119k + 82173k are squares for k = 1,2,3 (3642, 882702, 240811482).
30303k + 31213k + 34671k + 36309k are squares for k = 1,2,3 (3642, 664302, 121565082).
3642 = 132496, 1 + 3 + 2 + 4 + 9 + 6 = 52,
3642 = 132496, 1 + 3 + 2496 = 502,
3642 = 132496, 1 + 32 + 496 = 232.
272 + 282 + 292 + 302 + 312 + ... + 3642 = 40172,
2442 + 2452 + 2462 + 2472 + 2482 + ... + 3642 = 33662.
by Yoshio Mimura, Kobe, Japan
365
The smallest squares containing k 365's :
1236544 = 11122,
171365365369 = 4139632,
3650336536540225 = 604180152.
3652 = 133225, 1 + 3 + 3 + 2 + 2 + 5 = 42,
3652 = 133225, 13 + 33 + 33 + 23 + 23 + 53 = 142,
3652 = 133225, 13 + 3 + 2 + 2 + 5 = 52.
365 = 102 + 112 + 122 = 132 + 142.
13 - 23 + 33 - 43 + 53 - 63 + ... + 3633 - 3643 + 3653 = 49412.
3-by-3 magic squares consisting of different squares with constant 3652:
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Page of Squares : First Upload January 7, 2005 ; Last Revised February 16, 2009
by Yoshio Mimura, Kobe, Japan
366
The smallest squares containing k 366's :
153664 = 3922,
236636689 = 153832,
36643663666801 = 60534012.
240k + 306k + 313k + 366k are squares for k = 1,2,3 (352, 6192, 110532).
3662 = 1222 + 2442 + 2442 : 4422 + 4422 + 2212 = 6632.
3662 = 133956, a square with odd digits except the last digit 6.
Komachi Square Sum : 3662 = 212 + 672 + 852 + 3492.
3662 = 133956, 13 + 33 + 33 + 93 + 563 = 4202,
3662 = 133956, 13 + 3 + 9 + 5 + 6 = 62,
3662 = 133956, 13 + 3 + 9 + 56 = 92,
3662 = 133956, 13 + 3956 = 632,
3662 = 133956, 133 + 956 = 332.
(52 - 6)(92 - 6)(102 - 6) = 3662 - 6.
Page of Squares : First Upload January 7, 2005 ; Last Revised August 17, 2013by Yoshio Mimura, Kobe, Japan
367
The smallest squares containing k 367's :
367236 = 6062,
236736741136 = 4865562,
236733673367104 = 153861522.
3672 = 134689, a square with different digits.
3672 = 134689, a square with an increasing sequence of digits.
3672 = 24 + 44 + 84 + 194.
Komachi equation: 3672 = - 122 + 32 * 42 * 52 * 62 + 72 + 82 * 92.
Chain of squares :
3672 = 134689 -- 13689( = 1172) -- 1369( = 372) -- 169( = 132) -- 16( = 42) -- 1( = 12).
3-by-3 magic squares consisting of different squares with constant 3672:
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2942 + 2952 + 2962 + 2972 + 2982 + ... + 3672 = 28492.
Page of Squares : First Upload January 7, 2005 ; Last Revised June 1, 2010by Yoshio Mimura, Kobe, Japan
368
The smallest squares containing k 368's :
13689 = 1172,
3684368601 = 606992,
1368621368183689 = 369948832.
3682 = (12 + 7)(42 + 7)(272 + 7) = (32 + 7)(42 + 7)(192 + 7) = (52 + 7)(652 + 7).
3682 = 213 + 363 + 433 = 44 + 84 + 164 + 164.
Cubic Polynomial :
(X + 3682)(X + 5672)(X + 42842) = X3 + 43372X2 + 29032922X + 8938823042.
Komachi Fraction : 3682 = 7312896 / 54.
3682 = 135424, 13 + 33 + 53 + 43 + 23 + 43 = 172,
3682 = 135424, 1 + 3 + 54 + 2 + 4 = 82,
3682 = 135424, 1 + 35 + 4 + 24 = 82,
3682 = 135424, 1 + 354 + 2 + 4 = 192,
3682 = 135424, 13 + 5 + 42 + 4 = 82.
(12 + 22 + 32 + ... + 272)(282 + 292 + 302 + ... + 872)(882 + 892 + 902 + ... + 3682) = 1570846202.
3682 = 135424 appears in the decimal expression of e:
e = 2.71828•••135424••• (from the 121224th digit)
by Yoshio Mimura, Kobe, Japan
369
The smallest squares containing k 369's :
1369 = 372,
233692369 = 152872,
936952369369 = 9679632.
The squares which begin with 369 and end in 369 are
369139369 = 192132, 3695059369 = 607872, 36945837369 = 1922132,
36974290369 = 1922872, 369011296369 = 6074632,...
3692 = 136161, a square with 3 kinds of digits.
3692 = 136161, 1 * 3 + 6 * 1 * 61 = 1 + 3 + 61 * 6-1 = 1 * 3 + 61 * 6 * 1 = 1 + 361 + 6 + 1 = 369.
3692 = (52 + 2)(712 + 2).
3692 = 1232 + 2462 + 2462 : 6422 + 6422 + 3212 = 9632.
3692 = 104 + 104 + 154 + 164.
3-by-3 magic squares consisting of different squares with constant 3692:
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3692 = 136161, 13 + 6 + 1 + 61 = 92,
3692 = 136161, 13 + 6 + 16 + 1 = 62,
3692 = 136161, 13 + 61 + 6 + 1 = 92,
3692 = 136161, 136 + 1 + 6 + 1 = 122.
3692 = 136161 appears in the decimal expression of e:
e = 2.71828•••136161••• (from the 68372nd digit)
3692 = 136161, where 1 = 12, 36 = 62, 16 = 42.
Page of Squares : First Upload January 7, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan