## 5800

5800^{2}± 3 are primes.

5800^{2} = (1^{2} + 4)(2^{2} + 4)(11^{2} + 4)(82^{2} + 4) = (1^{2} + 4)(5^{2} + 4)(14^{2} + 4)(34^{2} + 4)

= (2^{2} + 4)(5^{2} + 4)(11^{2} + 4)(34^{2} + 4) = (6^{2} + 4)(11^{2} + 4)(82^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 5802

5802^{2} = 19^{4} + 41^{4} + 57^{4} + 67^{4} = 23^{4} + 41^{4} + 53^{4} + 69^{4}.

by Yoshio Mimura, Kobe, Japan

## 5804

5804^{2} = 134^{3} + 205^{3} + 283^{3}.

5804^{5} = 6586231911457025024 : 6^{2} + 5^{2} + 8^{2} + 6^{2} + 2^{2} + 3^{2} + 1^{2} + 9^{2} + 1^{2} + 1^{2} + 4^{2} + 5^{2} + 70^{2} + 2^{2} + 5^{2} + 0^{2} + 24^{2} = 5804.

by Yoshio Mimura, Kobe, Japan

## 5805

5805^{2} = 171^{3} + 243^{3} + 243^{3}.

by Yoshio Mimura, Kobe, Japan

## 5806

5806^{2}± 3 are primes.

5806^{2} = 33709636 appears in the decimal expressions of e:

e = 2.71828•••33709636••• (from the 101899th digit)

by Yoshio Mimura, Kobe, Japan

## 5807

5807^{2} = 9^{4} + 12^{4} + 24^{4} + 76^{4}.

by Yoshio Mimura, Kobe, Japan

## 5808

5808^{2} = 33732864, 3^{2} + 3^{2} + 7^{2} + 3^{2} + 2^{2} + 8^{2} + 6^{2} + 4^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 5809

5809 = (1^{2} + 2^{2} + 3^{2} + ... + 314^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 17^{2}).

by Yoshio Mimura, Kobe, Japan

## 5810

5810 = (1^{2} + 2^{2} + 3^{2} + ... + 332^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 18^{2}).

by Yoshio Mimura, Kobe, Japan

## 5811

1 / 5811 = 0.000172087420409...., 1^{2} + 72^{2} + 08^{2} + 7^{2} + 4^{2} + 20^{2} + 4^{2} + 09^{2} = 5811.

by Yoshio Mimura, Kobe, Japan

## 5814

5814^{2} = (10^{2} + 2)(13^{2} + 2)(44^{2} + 2) = (6^{2} + 2)(7^{2} + 2)(10^{2} + 2)(13^{2} + 2).

5814^{2} = (105 + 106 + 107 + ... + 116)^{2} + (117 + 118 + 119 + ... + 128)^{2} + (129 + 130 + 131 + ... + 140)^{2} + ... + (201 + 202 + 203 + ... + 212)^{2}.

5814^{4} = 1142615496339216, and 11^{2} + 42^{2} + 6^{2} + 1^{2} + 5^{2} + 49^{2} + 6^{2} + 33^{2} + 9^{2} + 2^{2} + 16^{2} = 5814.

by Yoshio Mimura, Kobe, Japan

## 5815

5815^{2} = 132^{3} + 201^{3} + 286^{3}.

1 / 5815 = 0.0001719690455717...., 17^{2} + 19^{2} + 69^{2} + 04^{2} + 5^{2} + 5^{2} + 7^{2} + 17^{2} = 5815.

by Yoshio Mimura, Kobe, Japan

## 5816

5816^{2} = 33825856, 3 * 3 * 8 * 2 * 5 * 8 + 56 = 5816.

5816^{4} = 1144188534132736, and 1^{2} + 1^{2} + 44^{2} + 1^{2} + 8^{2} + 8^{2} + 5^{2} + 3^{2} + 41^{2} + 3^{2} + 27^{2} + 36^{2} = 5816.

by Yoshio Mimura, Kobe, Japan

## 5819

5819^{2} = S_{2}(39) + S_{2}(466), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 5823

5823^{2} = 58^{3} + 86^{3} + 321^{3} = 81^{3} + 115^{3} + 317^{3}.

by Yoshio Mimura, Kobe, Japan

## 5825

5825^{2} = (11^{2} + 4)(521^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 5827

1 / 5827 = 0.00017161...., and 17161 = 131^{2}.

by Yoshio Mimura, Kobe, Japan

## 5828

5828^{5} = 6723535704575034368 : 6^{2} + 7^{2} + 23^{2} + 5^{2} + 3^{2} + 5^{2} + 7^{2} + 0^{2} + 4^{2} + 5^{2} + 7^{2} + 50^{2} + 34^{2} + 36^{2} + 8^{2} = 5828.

by Yoshio Mimura, Kobe, Japan

## 5830

S_{2}(5830) = S_{2}(3647) + S_{2}(5309), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 5831

5831^{2} = (32 + 33 + 34 + ... + 80)^{2} + (81 + 82 + 83 + ... + 129)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5832

5832^{2}± 5 are primes.

5832^{2} = 36^{3} + 216^{3} + 288^{3} = 39^{3} + 153^{3} + 312^{3} = 41^{3} + 114^{3} + 319^{3} = 45^{3} + 246^{3} + 267^{3}

= 72^{3} + 114^{3} + 318^{3} = 100^{3} + 192^{3} + 296^{3} = 102^{3} + 227^{3} + 277^{3} = 118^{3} + 186^{3} + 296^{3}

= 162^{3} + 216^{3} + 270^{3} = 173^{3} + 214^{3} + 267^{3} = 26^{4} + 26^{4} + 42^{4} + 74^{4} = 54^{4} + 54^{4} + 54^{4} + 54^{4}.

5832^{2} = (71 + 72 + 73 + ... + 94)^{2} + (95 + 96 + 97 + ... + 118)^{2} + (119 + 120 + 121 + ... + 142)^{2} + ... + (143 + 144 + 145 + ... + 166)^{2}.

Komachi equations:

5832^{2} = 9^{2} * 8^{2} / 7^{2} / 6^{2} * 54^{2} * 3^{2} * 21^{2},

5832^{2} = 98^{6} / 7^{6} / 6^{6} * 54^{6} * 3^{6} / 21^{6}.

by Yoshio Mimura, Kobe, Japan

## 5833

5833^{4} = 1157625022684321, and 11^{2} + 57^{2} + 6^{2} + 2^{2} + 5^{2} + 0^{2} + 2^{2} + 2^{2} + 6^{2} + 8^{2} + 43^{2} + 21^{2} = 5833.

by Yoshio Mimura, Kobe, Japan

## 5840

5840^{2} = 2^{6} + 6^{6} + 6^{6} + 18^{6}.

by Yoshio Mimura, Kobe, Japan

## 5841

5841^{2} = 34117281, 3 + 4 + 1 + 1 + 72 * 81 = 34 * 1 * 172 - 8 + 1 = 5841.

Komachi equation: 5841^{2} = 9^{6} - 8^{6} - 7^{6} - 6^{6} + 54^{6} / 3^{6} + 2^{6} + 1^{6}.

by Yoshio Mimura, Kobe, Japan

## 5843

5843^{2} = 34140649, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5844

5844^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 5845

5845^{3} = 199688726125, and 19^{2} + 9^{2} + 68^{2} + 8^{2} + 7^{2} + 2^{2} + 6^{2} + 1^{2} + 25^{2} = 5845.

5845^{5} = 6822170631552653125 : 6^{2} + 8^{2} + 2^{2} + 2^{2} + 1^{2} + 70^{2} + 6^{2} + 3^{2} + 1^{2} + 5^{2} + 5^{2} + 26^{2} + 5^{2} + 3^{2} + 1^{2} + 2^{2} + 5^{2} = 5845.

by Yoshio Mimura, Kobe, Japan

## 5846

5846^{2} = 34175716, a zigzag square.

5846^{2} = 34175716, 3 * 41 + 7 + 5716 = 5846.

by Yoshio Mimura, Kobe, Japan

## 5847

5847^{2} = 34^{3} + 257^{3} + 258^{3} = 14^{4} + 22^{4} + 54^{4} + 71^{4}.

by Yoshio Mimura, Kobe, Japan

## 5850

5850^{2} = (1^{2} + 9)(2^{2} + 9)(21^{2} + 9)(24^{2} + 9) = (1^{2} + 9)(2^{2} + 9)(3^{2} + 9)(4^{2} + 9)(24^{2} + 9)

= (1^{2} + 9)(2^{2} + 9)(4^{2} + 9)(6^{2} + 9)(15^{2} + 9) = (1^{2} + 9)(2^{2} + 9)(9^{2} + 9)(54^{2} + 9)

= (1^{2} + 9)(4^{2} + 9)(15^{2} + 9)(24^{2} + 9) = (1^{2} + 9)(6^{2} + 9)(11^{2} + 9)(24^{2} + 9)

= (2^{2} + 9)(3^{2} + 9)(4^{2} + 9)(6^{2} + 9)(11^{2} + 9) = (2^{2} + 9)(4^{2} + 9)(15^{2} + 9)(21^{2} + 9)

= (2^{2} + 9)(6^{2} + 9)(11^{2} + 9)(21^{2} + 9) = (2^{2} + 9)(9^{2} + 9)(171^{2} + 9)

= (3^{2} + 9)(4^{2} + 9)(11^{2} + 9)(24^{2} + 9) = (3^{2} + 9)(4^{2} + 9)(6^{2} + 9)(41^{2} + 9)

= (4^{2} + 9)(6^{2} + 9)(11^{2} + 9)(15^{2} + 9) = (6^{2} + 9)(21^{2} + 9)(41^{2} + 9) = (9^{2} + 9)(11^{2} + 9)(54^{2} + 9)

= (11^{2} + 9)(21^{2} + 9)(24^{2} + 9).

5850^{2} = (362 + 363 + 364)^{2} + (365 + 366 + 367)^{2} + (368 + 369 + 370)^{2} + (371 + 372 + 373)^{2} + ... + (431 + 432 + 433)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5853

5853^{2} = 34257609, a square consisting of just 3 kinds of digits.

5853^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 5856

5856^{2} = 34292736, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5859

5859^{2} = (2^{3} + 1)(6^{3} + 1)(26^{3} + 1).

5859^{2} = 54^{3} + 240^{3} + 273^{3}.

A cubic polynomial :

(X + 1440^{2})(X + 1612^{2})(X + 5859^{2}) = X^{3} + 6245^{2}X^{2} + 12875292^{2}X + 13600379520^{2}.

5859^{5} = 6904265602333993299 : 6^{2} + 9^{2} + 0^{2} + 42^{2} + 6^{2} + 5^{2} + 6^{2} + 0^{2} + 2^{2} + 33^{2} + 39^{2} + 9^{2} + 32^{2} + 9^{2} + 9^{2} = 5859.

by Yoshio Mimura, Kobe, Japan

## 5860

5860^{2}± 3 are primes.

5860^{2} = 156^{3} + 243^{3} + 253^{3} = 166^{3} + 190^{3} + 284^{3}.

Loop of length 14 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

5860 - 6964 - 8857 - 10993 - ... - 6505 - 4250 - 4264 - 5860

(Note f(5860) = 58^{2} + 60^{2} = 6964, f(6964) = 69^{2} + 64^{2} = 8857, etc. See 1946)

by Yoshio Mimura, Kobe, Japan

## 5863

5863^{2} = 34374769, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 5864

5864^{2} = 66^{3} + 110^{3} + 320^{3}.

by Yoshio Mimura, Kobe, Japan

## 5866

665^{k} + 1421^{k} + 2401^{k} + 5117^{k} are squares for k = 1,2,3 (98^{2}, 5866^{2}, 388570^{2}).

by Yoshio Mimura, Kobe, Japan

## 5868

5868^{2}± 5 are primes.

5868^{2} = 34433424, a square consisting of just 3 kinds of digits.

5868^{2} = (38 + 39 + 40)^{2} + (41 + 42 + 43)^{2} + (44 + 45 + 46)^{2} + ... + (323 + 324 + 325)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5871

5871^{2} = 97^{3} + 255^{3} + 257^{3}.

by Yoshio Mimura, Kobe, Japan

## 5872

5872^{2} = 8^{4} + 16^{4} + 32^{4} + 76^{4}.

1 / 5872 = 0.0001702997...., 1^{2} + 70^{2} + 29^{2} + 9^{2} + 7^{2} = 5872.

by Yoshio Mimura, Kobe, Japan

## 5874

5874 = (3^{2} + 2)(8^{2} + 2)(218^{2} + 2).

5874 = (1^{2} + 2^{2} + 3^{2} + ... + 44^{2}) / (1^{2} + 2^{2}).

5874^{5} = 6993099554303310624 : 6^{2} + 9^{2} + 9^{2} + 3^{2} + 0^{2} + 9^{2} + 9^{2} + 55^{2} + 43^{2} + 0^{2} + 3^{2} + 3^{2} + 1^{2} + 0^{2} + 6^{2} + 24^{2} = 5874.

by Yoshio Mimura, Kobe, Japan

## 5876

Loop of length 56 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

5876 - 9140 - 9881 - 16165 - ... - 6409 - 4177 - 7610 - 5876

(Note f(5876) = 58^{2} + 76^{2} = 9140, f(9140) = 91^{2} + 40^{2} = 9881, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 5878

5878^{2} = 171^{3} + 222^{3} + 265^{3}.

by Yoshio Mimura, Kobe, Japan

## 5879

5879^{2} = 1^{5} + 1^{5} + 12^{5} + 15^{5} + 32^{5}.

by Yoshio Mimura, Kobe, Japan

## 5880

5880^{2} = 31^{3} + 161^{3} + 312^{3}.

5880^{2} = 64^{3} + 65^{3} + 66^{3} + 67^{3} + 68^{3} + 69^{3} + ... + 111^{3}.

91^{2} + 5880 = 119^{2}, 91^{2} - 5880 = 49^{2}.

by Yoshio Mimura, Kobe, Japan

## 5883

1 / N = 0.000169...., and 169 = 13^{2} for N = 5883, 5884,..., 5917.

5883^{2} = 34609689, 34 + 60 * 96 + 89 = 5883.

by Yoshio Mimura, Kobe, Japan

## 5885

5885^{5} = 7058823711905178125 : 7^{2} + 0^{2} + 5^{2} + 8^{2} + 8^{2} + 2^{2} + 3^{2} + 71^{2} + 1^{2} + 9^{2} + 0^{2} + 5^{2} + 17^{2} + 8^{2} + 12^{2} + 5^{2} = 70^{2} + 5^{2} + 8^{2} + 8^{2} + 2^{2} + 3^{2} + 7^{2} + 1^{2} + 19^{2} + 0^{2} + 5^{2} + 17^{2} + 8^{2} + 1^{2} + 2^{2} + 5^{2} = 5885.

1276^{k} + 1529^{k} + 1738^{k} + 5258^{k} are squares for k = 1,2,3 (99^{2}, 5885^{2}, 395307^{2}).

by Yoshio Mimura, Kobe, Japan

## 5886

5886^{2}± 5 are primes.

5886^{2} = 107^{3} + 145^{3} + 312^{3}.

5886^{2} + 5887^{2} + 5888^{2} + ... + 5940^{2} = 5941^{2} + 5942^{2} + 5943^{2} + ... + 5994^{2}.

5886^{5} = 7064823051786334176 : 7^{2} + 0^{2} + 64^{2} + 8^{2} + 2^{2} + 30^{2} + 5^{2} + 17^{2} + 8^{2} + 6^{2} + 3^{2} + 3^{2} + 4^{2} + 17^{2} + 6^{2} = 5886.

by Yoshio Mimura, Kobe, Japan

## 5887

5887^{2} = 29^{4} + 58^{4} + 58^{4} + 58^{4}.

by Yoshio Mimura, Kobe, Japan

## 5888

5888^{2} = (1^{2} + 7)(3^{2} + 7)(19^{2} + 7)(27^{2} + 7) = (1^{2} + 7)(3^{2} + 7)(4^{2} + 7)(5^{2} + 7)(19^{2} + 7)

= (3^{2} + 7)(4^{2} + 7)(11^{2} + 7)(27^{2} + 7) = (4^{2} + 7)(5^{2} + 7)(11^{2} + 7)(19^{2} + 7)

= (11^{2} + 7)(19^{2} + 7)(27^{2} + 7).

5888^{2} = 16^{4} + 32^{4} + 64^{4} + 64^{4} = 8^{5} + 8^{5} + 16^{5} + 32^{5}.

The square root of 5888 is 76.7333...., and 76 = 7^{2} + 3^{2} + 3^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 5894

5894^{2} = 127^{3} + 245^{3} + 262^{3}.

by Yoshio Mimura, Kobe, Japan

## 5895

5895^{2} = (200 + 201 + 202)^{2} + (203 + 204 + 205)^{2} + (206 + 207 + 208)^{2} + (209 + 210 + 211)^{2} + ... + (347 + 348 + 349)^{2}.

by Yoshio Mimura, Kobe, Japan

## 5896

5896^{2} = (1^{2} + 7)(23^{2} + 7)(90^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 5897

5897^{2} = 4^{4} + 52^{4} + 52^{4} + 67^{4}.

5897^{2} = 34774609, 3^{2} + 4^{2} + 7^{2} + 7^{2} + 4^{2} + 6^{2} + 0^{2} + 9^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 5898

5898^{2} = 34786404, 3 / 4 * 7864 + 0 * 4 = 5898.

by Yoshio Mimura, Kobe, Japan

## 5899

5899^{2} = 34798201, a square with different digits.

by Yoshio Mimura, Kobe, Japan