2300
23002 = 104 + 204 + 404 + 404.
23002 = (33 + 34 + 35 + 36 + 37)2 + (38 + 39 + 40 + 41 + 42)2 + (43 + 44 + 45 + 46 + 47)2 + ... + (143 + 144 + 145 + 146 + 147)2.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2301
23012 = 5294601, a zigzag square with different digits.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2304
The square of 48.
23042 = 5308416, a square with different digits.
2304 = 482, a zigzag square with different digits.
23042 = 603 + 883 + 1643 = 803 + 1003 + 1563.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2305
23052 = 1083 + 1243 + 1293.
1081 + 1241 + 1291 = 192, 1082 + 1242 + 1292 = 2092, 1083 + 1243 + 1293 = 23052 (See 19).
Page of Squares : First Upload July 22, 2008 ; Last Revised January 11, 2011by Yoshio Mimura, Kobe, Japan
2309
23092 = 923 + 1293 + 1343.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2310
924k + 1078k + 1617k + 2310k are squares for k = 1,2,3 (772, 31572, 1363672).
23102 = (12 + 6)(182 + 6)(482 + 6) = (12 + 6)(22 + 6)(152 + 6)(182 + 6)
= (12 + 6)(22 + 6)(32 + 6)(42 + 6)(152 + 6) = (12 + 6)(32 + 6)(42 + 6)(482 + 6)
= (12 + 6)(32 + 6)(42 + 6)(62 + 6)(72 + 6) = (12 + 6)(42 + 6)(122 + 6)(152 + 6)
= (12 + 6)(62 + 6)(72 + 6)(182 + 6) = (22 + 6)(152 + 6)(482 + 6)
= (22 + 6)(32 + 6)(62 + 6)(292 + 6) = (22 + 6)(62 + 6)(72 + 6)(152 + 6)
= (32 + 6)(42 + 6)(82 + 6)(152 + 6) = (32 + 6)(62 + 6)(922 + 6)
= (42 + 6)(182 + 6)(272 + 6) = (62 + 6)(122 + 6)(292 + 6)
= (62 + 6)(72 + 6)(482 + 6) = (82 + 6)(152 + 6)(182 + 6).
The integral triangle of sides 4474, 6273, 10549 (or 451, 27608, 27843) has square area 23102.
Page of Squares : First Upload May 17, 2011 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2311
23112 = 5340721, a square with different digits.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2312
23122 = 5345344, a square with just 3 kinds of squares.
Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
2312 - 673 - 5365 - 7034 - ... - 2797 - 10138 - 1446 - 2312
(Note f(2312) = 232 + 122 = 673, f(673) = 62 + 732 = 5365, etc. See 37)
23122 = 683 + 1363 + 1363 = 344 + 344 + 344 + 344.
173 + 2312 = 852, 173 - 2312 = 512.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 27, 2011by Yoshio Mimura, Kobe, Japan
2313
23132 = 93 + 1123 + 1583 = 34 + 124 + 124 + 484.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2315
23152 = 47 + 67 + 67 + 97.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2316
23162 = 5363856, a zigzag square.
23162 = 223 + 443 + 1743.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2320
23202 = (12 + 4)(22 + 4)(42 + 4)(822 + 4) = (22 + 4)(242 + 4)(342 + 4)
= (22 + 4)(42 + 4)(52 + 4)(342 + 4) = (22 + 4)(52 + 4)(62 + 4)(242 + 4)
= (42 + 4)(62 + 4)(822 + 4).
by Yoshio Mimura, Kobe, Japan
2321
23212 = 5387041, a square with different digits.
Komachi equation: 23212 = 14 + 24 - 344 + 54 - 674 + 84 * 94.
23212 = 493 + 1343 + 1423.
Page of Squares : First Upload April 2, 2007 ; Last Revised September 21, 2010by Yoshio Mimura, Kobe, Japan
2322
23222 = 5391684, a square with different digits.
23222± 5 are primes.
23222 = (12 + 5)(22 + 5)(92 + 5)(342 + 5) = (22 + 2)(162 + 2)(592 + 2) = (72 + 5)(92 + 5)(342 + 5).
Page of Squares : First Upload April 2, 2007 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
2323
23232 = 30 + 33 + 35 + 36 + 37 + 39 + 310 + 312 + 314.
Page of Squares : First Upload August 29, 2011 ; Last Revised August 29, 2011by Yoshio Mimura, Kobe, Japan
2324
23242± 3 are primes.
Page of Squares : First Upload January 16, 2014 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
2325
23252 = 153 + 353 + 1753.
430k + 740k + 1470k + 1585k are squares for k = 1,2,3 (652, 23252, 874252).
Page of Squares : First Upload July 22, 2008 ; Last Revised May 17, 2011by Yoshio Mimura, Kobe, Japan
2326
23262 = 5410276, a square with different digits.
23262 = 5410276, and 5 * 410 + 276 = 2326.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2327
23272 = 563 + 1133 + 1563.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2329
23292 = 5424241, a square pegged by 4.
23292 = (80 + 81 + 82 + ... + 96)2 + (97 + 98 + 99 + ... + 113)2 + (114 + 115 + 116 + ... + 130)2 + ... + (97 + 98 + 99 + ... + 113)2.
2329k + 38908k + 53978k + 73706k are squares for k = 1,2,3 (4112, 993252, 248313872).
Page of Squares : First Upload April 2, 2007 ; Last Revised May 17, 2011by Yoshio Mimura, Kobe, Japan
2330
23302 = (42 + 4)(5212 + 4).
Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2331
23312 = 13 - 23 + 33 - 43 + 53 - 63 + ... + 2213.
23312 = 5433561, 52 + 42 + 32 + 32 + 52 + 62 + 12 = 112.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2332
23322 = 84 + 124 + 184 + 484.
23322 = (22 + 7)(242 + 7)(292 + 7).
Page of Squares : First Upload July 22, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2334
23342 = 5447556, and 5 * 4 + 4 + 7 * 55 * 6 = 2334.
762k + 1938k + 2334k + 3066k are squares for k = 1,2,3 (902, 43802, 2219402).
Page of Squares : First Upload April 2, 2007 ; Last Revised May 17, 2011by Yoshio Mimura, Kobe, Japan
2335
23352 = 5452225, a square with just 3 kinds of digits.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2337
23372 = 353 + 393 + 1753 = 643 + 1373 + 1383 = 84 + 164 + 174 + 484.
1 / 2337 = 0.00042789901583, 422 + 72 + 82 + 92 + 92 + 0152 + 82 + 32 = 2337.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2338
994k + 1470k + 2338k + 4802k are squares for k = 1,2,3 (982, 56282, 3573082).
Page of Squares : First Upload May 17, 2011 ; Last Revised May 17, 2011by Yoshio Mimura, Kobe, Japan
2339
23392 = 5470921, a square with different digits.
23392 = 5470921, and 5 * 470 - 9 - 2 * 1 = 5 * 470 - 9 - 2 / 1 = 2339.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2340
23402 = (12 + 9)(32 + 9)(112 + 9)(152 + 9) = (22 - 1)(13512 - 1) = (22 - 1)(42 - 1)(142 - 1)(252 - 1).
Komachi equations:
23402 = 122 * 32 / 42 + 52 * 62 * 782 - 92 = 122 * 32 / 42 * 52 * 62 * 782 / 92
= 122 * 32 * 452 / 62 * 782 / 92 = - 122 * 32 / 42 + 52 * 62 * 782 + 92.
23402 = 133 + 593 + 1743 = 333 + 1363 + 1433.
Page of Squares : First Upload July 22, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2343
23432 = 5489649 appears in the decimal expressions of π:
π = 3.14159•••5489649••• (from the 16513rd digit)
(5489649 is the fourth 7-digit square in the expression of π.)
by Yoshio Mimura, Kobe, Japan
2344
23442 = 12 + 92 + 172 + ... + (8x + 1)2 + ... + 5052.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2346
23462 + 23472 + 23482 + ... + 23802 = 23812 + 23822 + 23832 + ... + 24142.
Page of Squares : First Upload September 13, 2011 ; Last Revised September 13, 2011by Yoshio Mimura, Kobe, Japan
2347
23475 = 71214011112376507 : 72 + 12 + 22 + 12 + 402 + 12 + 12 + 12 + 12 + 232 + 72 + 62 + 52 + 02 + 72 = 2347.
Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
2349
23492 = (176 + 177 + 178 + 179 + 180 + 181 + 182 + 183 + 184)2 + (185 + 186 + 187 + 188 + 189 + 190 + 191 + 192 + 193)2.
23492 = 813 + 1263 + 1443.
23494 = 30446127875601, and 32 + 02 + 442 + 62 + 122 + 72 + 82 + 72 + 52 + 62 + 02 + 12 = 2349.
Page of Squares : First Upload April 2, 2007 ; Last Revised December 1, 2008by Yoshio Mimura, Kobe, Japan
2350
23502 = 5522500 appears in the decimal expressions of π:
π = 3.14159•••5522500••• (from the 82332nd digit)
by Yoshio Mimura, Kobe, Japan
2351
2351 is the 5th prime for which the Lendre symbol (a/2351) = 1 for a = 1, 2, ..., 12.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2352
23522 = 143 + 1333 + 1473 = 703 + 813 + 1673.
23522 = (12 + 3)(32 + 3)(92 + 3)(372 + 3).
Page of Squares : First Upload July 22, 2008 ; Last Revised December 21, 2013by Yoshio Mimura, Kobe, Japan
2353
481k + 624k + 1404k + 1716k are squares for k = 1,2,3 (652, 23532, 904152).
Page of Squares : First Upload May 17, 2011 ; Last Revised May 17, 2011by Yoshio Mimura, Kobe, Japan
2356
23562± 3 are primes.
1 / 2356 = 0.000424448, 42 + 22 + 42 + 42 + 482 = 2356.
23562 = 5550736, 52 + 52 + 52 + 02 + 72 + 32 + 62 = 132.
1488k + 2356k + 9145k + 11036k are squares for k = 1,2,3 (1552, 146012, 14578372).
Page of Squares : First Upload April 2, 2007 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
2357
23572 = 5555449, a square with just 3 kinds of digits.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2358
1 / 2358 = 0.0004240882103, 422 + 42 + 02 + 82 + 82 + 212 + 02 + 32 = 2358,
1 / 2358 = 0.0004240882103, 422 + 42 + 02 + 82 + 82 + 212 + 032 = 2358,
1 / 2358 = 0.0004240882103, 422 + 42 + 082 + 82 + 212 + 02 + 32 = 2358,
1 / 2358 = 0.0004240882103, 422 + 42 + 082 + 82 + 212 + 032 = 2358.
by Yoshio Mimura, Kobe, Japan
2359
23592 = 5564881, and 55 + 6 * 48 * 8 * 1 = 55 + 6 * 48 * 8 / 1 = 2359.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2363
12 + 22 + 32 + 42 + ... + 23632 = 4400940994, the sum with just 3 kinds of digits.
S2(2363) = S2(1058) + S2(2290), where Sn = 12 + 22 + 32 + 42 + ... + n2.
23632 = 5583769, 52 + 52 + 82 + 32 + 72 + 62 + 92 = 172.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2364
23642± 5 are primes.
Page of Squares : First Upload April 2, 2007 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
2366
23662 = 5597956, a square with odd digits except the last digit 6.
23662± 3 are primes.
23662 = 263 + 913 + 1693.
23662 = (192 + 3)(1242 + 3) = (22 + 3)(72 + 3)(1242 + 3).
Page of Squares : First Upload July 22, 2008 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
2367
23672 = 93 + 1063 + 1643 = 1203 + 1203 + 1293.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2369
23692 = 5612161, a zigzag square.
23695 = 74614855732022849 : 72 + 42 + 62 + 142 + 82 + 52 + 52 + 72 + 322 + 02 + 22 + 282 + 42 + 92 = 2369.
Page of Squares : First Upload April 2, 2007 ; Last Revised December 8, 2008by Yoshio Mimura, Kobe, Japan
2370
23702 = 54 x 55 + 56 x 57 + 58 x 59 + 60 x 61 + 62 x 63 + ... + 322 x 323.
23702 = 5616900 appears in the decimal expressions of e:
e = 2.71828•••5616900••• (from the 58308th digit)
by Yoshio Mimura, Kobe, Japan
2372
23722 = 5626384, a zigzag square.
23722 = 5626384, and 5 * 6 * 26 * 3 + 8 * 4 = 2372.
23722 = 1932 + 1942 + 1952 + 1962 + 1972 + 1982 + ... + 2882.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2376
23762 = 5645376, a zigzag square.
23762 = 5645376, 52 + 62 + 42 + 52 + 32 + 72 + 62 = 142.
23762 = (12 + 8)(42 + 8)(52 + 8)(282 + 8) = (12 + 8)(52 + 8)(82 + 8)(162 + 8)
= (22 - 1)(32 - 1)(4852 - 1) = (42 + 8)(172 + 8)(282 + 8) = (52 - 1)(4852 - 1)
= (82 + 8)(162 + 8)(172 + 8) = (82 + 8)(2802 + 8).
by Yoshio Mimura, Kobe, Japan
2379
23792 = 4542 + 4552 + 4562 + 4572 + 4582 + 4592 + ... + 4792.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2381
A cubic polynomial :
(X + 9122)(X + 12112)(X + 18362) = X3 + 23812X2 + 29944922X + 20277371522.
by Yoshio Mimura, Kobe, Japan
2382
23822 = 5673924, a square with different digits.
23822 = 1033 + 1253 + 1383.
Page of Squares : First Upload April 2, 2007 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2383
23832 = 34 + 144 + 244 + 484.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2384
23842 = 303 + 1123 + 1623.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2388
23882 = 503 + 1143 + 1603.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2394
The integral triangle of sides 399, 41210, 41497 has square area 23942.
Komachi equation: 23942 = 12 * 22 / 32 * 4562 * 72 / 82 * 92.
Page of Squares : First Upload July 22, 2008 ; Last Revised October 7, 2011by Yoshio Mimura, Kobe, Japan
2395
23952 = 73 + 73 + 1793.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan
2396
23962 = 5740816, a different digits.
Page of Squares : First Upload April 2, 2007 ; Last Revised April 2, 2007by Yoshio Mimura, Kobe, Japan
2397
23972 = 253 + 1403 + 1443.
Page of Squares : First Upload July 22, 2008 ; Last Revised July 22, 2008by Yoshio Mimura, Kobe, Japan