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590

The smallest squares containing k 590's :
59049 = 243²,
105908590096 = 325436²,
990590959059025 = 31473655².

590² + 591² + 592² + 593² + ... + 807² = 10355²,
590² + 591² + 592² + 593² + ... + 886² = 12804²,
590² + 591² + 592² + 593² + ... + 2328² = 64343².

113^k + 124^k + 262^k + 590^k are squares for k = 1,2,3 (33², 667², 15057²).
254^k + 362^k + 394^k + 590^k are squares for k = 1,2,3 (40², 836²,18176²).
3245^k + 7965^k + 101185^k + 235705^k are squares for k = 1,2,3 (590², 256650²,118876150²).

Page of Squares : First Upload June 6, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

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591

The smallest squares containing k 591's :
591361 = 769²,
459159184 = 21428²,
1591105914591561 = 39888669².

1 / 591 = 0.00169..., 169 = 13².

591² = 349281, a square with different digits.

(1³ + 2³ + ... + 255³)(256³ + 257³ + ... + 591³) = 5609640960².

3-by-3 magic squares consisting of different squares with constant 591²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(9, 132, 576, 288, 504, 111, 516, 279, 72),	(14, 242, 539, 286, 469, 218, 517, 266, 106),
(22, 106, 581, 139, 566, 98, 574, 133, 46),	(26, 181, 562, 274, 502, 149, 523, 254, 106),
(26, 254, 533, 341, 442, 194, 482, 299, 166),	(34, 203, 554, 358, 434, 181, 469, 346, 98),
(37, 166, 566, 386, 422, 149, 446, 379, 82),	(43, 134, 574, 266, 518, 101, 526, 251, 98),
(46, 238, 539, 406, 379, 202, 427, 386, 134),	(50, 166, 565, 334, 475, 110, 485, 310, 134),
(72, 324, 489, 369, 408, 216, 456, 279, 252),	(86, 187, 554, 229, 526, 142, 538, 194, 149),
(86, 293, 506, 331, 446, 202, 482, 254, 229),	(98, 314, 491, 389, 406, 182, 434, 293, 274),
(106, 331, 478, 373, 334, 314, 446, 358, 149),	(149, 274, 502, 358, 434, 181, 446, 293, 254)

591² = 349281, 3 + 49 + 28 + 1 = 9²,
591² = 349281, 34 + 9 + 281 = 18²,
591² = 349281, 3 + 492 + 81 = 24²,
591² = 349281, 3 + 49281 = 222².

Page of Squares : First Upload June 6, 2005 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

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592

The smallest squares containing k 592's :
5929 = 77²,
59259204 = 7698²,
255925925359225 = 15997685².

592² + 593² + 594² + 595² + ... + 2535² = 73242².

592² = 350464, 35 + 0 + 4 + 6 + 4 = 7²,
592² = 350464, 3 + 50 + 4 + 64 = 11²,
592² = 350464, 350 + 46 + 4 = 20².

Page of Squares : First Upload June 6, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

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593

The smallest squares containing k 593's :
295936 = 544²,
2593559329 = 50927²,
1593432593059329 = 39917823².

593² = 351649, a zigzag square with different digits.

3-by-3 magic squares consisting of different squares with constant 593²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(8, 183, 564, 393, 424, 132, 444, 372, 127),	(24, 73, 588, 172, 564, 63, 567, 168, 44),
(36, 312, 503, 388, 393, 216, 447, 316, 228),	(48, 332, 489, 361, 372, 288, 468, 321, 172),
(60, 255, 532, 343, 420, 240, 480, 332, 105),	(63, 348, 476, 388, 336, 297, 444, 343, 192),
(73, 204, 552, 372, 447, 116, 456, 332, 183)

593² = 351649, 3 + 5 + 1 + 6 + 49 = 8²,
593² = 351649, 35 + 16 + 4 + 9 = 8²,
593² = 351649, 35 + 16 + 49 = 10²,
593² = 351649, 3³ + 5³ + 16³ + 4³ + 9³ = 71².

Page of Squares : First Upload June 6, 2005 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

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594

The smallest squares containing k 594's :
594441 = 771²,
9594594304 = 97952²,
13594594335947881 = 116595859².

594² = 352836, a zigzag square.

594²± 5 are primes.

594² = (1² + 2)(2² + 2)(140² + 2) = (1² + 2)(2² + 2)(3² + 2)(5² + 2)(8² + 2)
= (2² + 2)(3² + 2)(5² + 2)(14² + 2) = (3² + 2)(4² + 2)(5² + 2)(8² + 2)
= (3² + 2)(8² + 2)(22² + 2) = (4² + 2)(140² + 2) = (5² + 2)(8² + 2)(14² + 2).

(1 + 2 + 3)(4 + 5 + 6 + 7)(8 + 9 + ... + 73) = 594²,
(1 + 2)(3 + 4 + 5 + 6)(7 + 8 + ... + 114) = 594².

Komachi equations:
594² = - 1⁵ + 2⁵ + 3⁵ * 4⁵ + 5⁵ - 6⁵ + 7⁵ + 8⁵ + 9⁵ = 9⁵ + 8⁵ + 7⁵ - 6⁵ + 5⁵ + 4⁵ * 3⁵ + 2⁵ - 1⁵.

The 4-by-4 magic square consisting of different squares with constant 594:

594² = 352836, 35 + 2 + 8 + 36 = 9²,
594² = 352836, 3 + 52 + 83 + 6 = 12²,
594² = 352836, 35 + 283 + 6 = 18²,
594² = 352836, 352 + 83 + 6 = 21².

Page of Squares : First Upload June 6, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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595

The smallest squares containing k 595's :
59536 = 244²,
9559559529 = 97773²,
89595595595025 = 9465495².

595 = (1² + 2² + 3² + ... + 399²) / (1² + 2² + 3² + ... + 47²).

595² = 42³ + 1⁵ + 6⁷.

595² + 596² + 597² + ... + 612² = 613² + 614² + 615² + ... + 629².

Komachi square sums : 595² = 14² + 32² + 97² + 586² = 8² + 91² + 376² + 452².

3-by-3 magic squares consisting of different squares with constant 595²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 280, 525, 357, 420, 224, 476, 315, 168),	(10, 198, 561, 330, 465, 170, 495, 314, 102),
(15, 170, 570, 262, 510, 159, 534, 255, 62),	(15, 170, 570, 270, 510, 145, 530, 255, 90),
(15, 206, 558, 270, 495, 190, 530, 258, 81),	(15, 270, 530, 354, 422, 225, 478, 321, 150),
(18, 390, 449, 415, 330, 270, 426, 305, 282),	(30, 150, 575, 305, 498, 114, 510, 289, 102),
(30, 225, 550, 370, 438, 159, 465, 334, 162),	(30, 305, 510, 370, 390, 255, 465, 330, 170),
(46, 303, 510, 390, 370, 255, 447, 354, 170),	(90, 226, 543, 255, 510, 170, 530, 207, 174),
(90, 255, 530, 369, 442, 150, 458, 306, 225),	(102, 289, 510, 414, 402, 145, 415, 330, 270),
(129, 278, 510, 330, 465, 170, 478, 246, 255)

595² = 354025, 3 + 54 + 0 + 2 + 5 = 8²,
595² = 354025, 35 + 4 + 0 + 25 = 8²,
595² = 354025, 35 + 40 + 25 = 10²,
595² = 354025, 354 + 0 + 2 + 5 = 19²,
595² = 354025, 35³ + 4³ + 0³ + 25³ = 242²,
595² = 354025, 35³ + 40³ + 25³ = 350².

Page of Squares : First Upload June 6, 2005 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

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596

The smallest squares containing k 596's :
34596 = 186²,
3059638596 = 55314²,
175965159614596 = 13265186².

The squares which begin with 596 and end in 596 are
59626802596 = 244186²,   59689330596 = 244314²,   596271218596 = 772186²,
596468914596 = 772314²,   5960014046596 = 2441314²,...

596² = 355216, 35 + 5 + 2 + 1 + 6 = 7²,
596² = 355216, 3² + 5² + 5² + 2² + 1² + 6² = 10²,
596² = 355216, 35 + 5 + 216 = 16².

596² = 355216 appears in the decimal expression of e:
  e = 2.71828•••355216••• (from the 101323rd digit).

Page of Squares : First Upload January 16, 2006 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

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597

The smallest squares containing k 597's :
597529 = 773²,
37597597801 = 193901²,
5975975979886969 = 77304437².

597² = 356409, a square with different digits.

597² = 17³ + 40³ + 66³.

3-by-3 magic squares consisting of different squares with constant 597²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 77, 592, 307, 508, 64, 512, 304, 43),	(13, 76, 592, 332, 493, 56, 496, 328, 53),
(18, 186, 567, 402, 423, 126, 441, 378, 138),	(20, 365, 472, 403, 340, 280, 440, 328, 235),
(28, 172, 571, 389, 428, 148, 452, 379, 92),	(32, 104, 587, 196, 557, 88, 563, 188, 64),
(32, 181, 568, 269, 512, 148, 532, 248, 109),	(53, 112, 584, 176, 563, 92, 568, 164, 83),
(53, 200, 560, 400, 428, 115, 440, 365, 172),	(59, 308, 508, 412, 389, 188, 428, 332, 251),
(78, 279, 522, 414, 402, 153, 423, 342, 246),	(83, 356, 472, 412, 307, 304, 424, 368, 203),
(88, 301, 508, 328, 452, 211, 491, 248, 232),	(92, 197, 556, 284, 508, 133, 517, 244, 172)

597² = 356409, 3 + 5 + 64 + 0 + 9 = 9²,
597² = 356409, 3 + 564 + 0 + 9 = 24².

597² = 356409 appears in the decimal expression of e:
  e = 2.71828•••356409••• (from the 61250th digit).

Page of Squares : First Upload June 6, 2005 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

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598

The smallest squares containing k 598's :
595984 = 772²,
3059859856 = 55316²,
5987598559849 = 2446957².

598² = 357604, a square with different digits.

(1³ + 2³ + ... + 7³)(8³ + 9³ + ... + 115³)(116³ + 117³ + ... + 598³) = 33425404488².

598² = 357604, 3 + 5 + 7 + 6 + 0 + 4 = 5²,
598² = 357604, 357 + 604 = 31².

598² = 357604 appears in the decimal expression of e:
  e = 2.71828•••357604••• (from the 8189th digit),
  (357604 is the sixth 6-digit square in the expression of e.)

Page of Squares : First Upload June 6, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

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599

The smallest squares containing k 599's :
599076 = 774²,
59940259929 = 244827²,
1599259925599504 = 39990748².

599 is the 10th prime for which the Legendere symbol (n/p) = 1 for n = 1, 2,..., 6.

3-by-3 magic squares consisting of different squares with constant 599²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 77, 594, 418, 426, 51, 429, 414, 58),	(13, 246, 546, 366, 429, 202, 474, 338, 141),
(14, 141, 582, 174, 558, 131, 573, 166, 54),	(14, 354, 483, 381, 378, 266, 462, 301, 234),
(30, 285, 526, 365, 426, 210, 474, 310, 195),	(42, 274, 531, 419, 366, 222, 426, 387, 166),
(51, 222, 554, 282, 499, 174, 526, 246, 147),	(54, 211, 558, 387, 414, 194, 454, 378, 99),
(78, 266, 531, 306, 477, 194, 509, 246, 198),	(90, 274, 525, 355, 450, 174, 474, 285, 230),
(131, 306, 498, 342, 454, 189, 474, 243, 274)

599² = 358801, 3 + 5 + 8 + 8 + 0 + 1 = 5²,
599² = 358801, 35 + 8801 = 94².

Page of Squares : First Upload January 16, 2006 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

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