## 6000

6000^{2} = 100^{3} + 200^{3} + 300^{3}.

85^{2} + 6000 = 115^{2}, 85^{2} - 6000 = 35^{2}.

6000^{2} = 36000000 appears in the decimal expressions of e:

e = 2.71828•••36000000••• (from the 89294th digit)

by Yoshio Mimura, Kobe, Japan

## 6001

6001^{2} = 199^{2} + 200^{2} + 201^{2} + 202^{2} + ... + 487^{2}.

by Yoshio Mimura, Kobe, Japan

## 6002

6002^{2} = 36024004, 3^{2} + 6^{2} + 0^{2} + 2^{2} + 4^{2} + 0^{2} + 0^{2} + 4^{2} = 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 6003

6003^{2} = 36036009, 3 - 6 + 0 - 3 + 6009 = 6003.

52^{k} + 190^{k} + 233^{k} + 254^{k} are squares for k = 1,2,3 (27^{2}, 397^{2}, 6003^{2}).

by Yoshio Mimura, Kobe, Japan

## 6005

6005^{2} = 36060025, 36 = 6^{2} and 60025 = 245^{2}.

6005^{5} = 7808454045018753125 :

7^{2} + 8^{2} + 0^{2} + 8^{2} + 45^{2} + 40^{2} + 45^{2} + 0^{2} + 1^{2} + 8^{2} + 7^{2} + 5^{2} + 3^{2} + 1^{2} + 2^{2} + 5^{2} = 6005.

by Yoshio Mimura, Kobe, Japan

## 6006

6006^{2} = 39^{3} + 206^{3} + 301^{3}.

The integral triangle of sides 3528, 20735, 21601 (or 5626, 66297, 71825) has square area 6006^{2}.

by Yoshio Mimura, Kobe, Japan

## 6009

1 / 6009 = 0.00016641...., and 16641 = 129^{2}.

by Yoshio Mimura, Kobe, Japan

## 6011

1 / 6011 = 0.00016636167027...., 1^{2} + 66^{2} + 36^{2} + 16^{2} + 7^{2} + 0^{2} + 2^{2} + 7^{2} = 6011,

1 / 6011 = 0.00016636167027...., 1^{2} + 66^{2} + 36^{2} + 16^{2} + 7^{2} + 02^{2} + 7^{2} = 6011.

by Yoshio Mimura, Kobe, Japan

## 6012

6012^{2} = 118^{3} + 183^{3} + 305^{3}.

by Yoshio Mimura, Kobe, Japan

## 6013

6013^{2} = 36156169, where 361 = 19^{2} and 56169 = 237^{2}.

6013^{2} = 36156169, 3^{2} + 6^{2} + 1^{2} + 5^{2} + 6^{2} + 1^{2} + 6^{2} + 9^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 6014

6014^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 6016

6016^{2} = 36192256, 3^{2} + 6^{2} + 1^{2} + 9^{2} + 2^{2} + 2^{2} + 5^{2} + 6^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 6017

6017^{2} = 48^{4} + 48^{4} + 56^{4} + 63^{4}.

by Yoshio Mimura, Kobe, Japan

## 6020

6020^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 6022

6022^{2} = 185^{3} + 187^{3} + 286^{3}.

by Yoshio Mimura, Kobe, Japan

## 6024

6024^{2} = 156^{3} + 182^{3} + 298^{3}.

by Yoshio Mimura, Kobe, Japan

## 6025

6025^{2} = 36300625, 3 * 6 * 300 + 625 = 3 * 6 + 3006 * 2 - 5 = 36300 / 6 - 25 = 6025.

by Yoshio Mimura, Kobe, Japan

## 6027

6027^{4} = 1319485936923441,

and 1^{2} + 31^{2} + 9^{2} + 4^{2} + 8^{2} + 59^{2} + 36^{2} + 9^{2} + 2^{2} + 3^{2} + 4^{2} + 4^{2} + 1^{2} = 6027,

6027^{4} = 1319485936923441,

and 13^{2} + 19^{2} + 48^{2} + 5^{2} + 9^{2} + 36^{2} + 9^{2} + 2^{2} + 3^{2} + 4^{2} + 41^{2} = 6027.

by Yoshio Mimura, Kobe, Japan

## 6030

6030^{2} = 36360900, 3 - 63 + 6090 + 0 = 6030.

Komachi equation: 6030^{2} = 12^{2} / 3^{2} * 4^{2} * 5^{2} * 67^{2} / 8^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 6031

6031^{2} = 9^{3} + 144^{3} + 322^{3}.

6031^{2} = 36372961, 3^{2} + 6^{2} + 3^{2} + 7^{2} + 2^{2} + 9^{2} + 6^{2} + 1^{2} = 15^{2}.

by Yoshio Mimura, Kobe, Japan

## 6032

6032^{4} = 1323869971480576,

and 1^{2} + 3^{2} + 23^{2} + 8^{2} + 6^{2} + 9^{2} + 9^{2} + 71^{2} + 4^{2} + 8^{2} + 0^{2} + 5^{2} + 7^{2} + 6^{2} = 6032.

by Yoshio Mimura, Kobe, Japan

## 6034

6034^{2} = 5^{4} + 29^{4} + 65^{4} + 65^{4}.

by Yoshio Mimura, Kobe, Japan

## 6037

6037^{2} = 36445369 appears in the decimal expressions of π:

π = 3.14159•••36445369••• (from the 11258th digit)

(36445369 is the first 8-digit square in the expression of π.)

by Yoshio Mimura, Kobe, Japan

## 6038

6038^{4} = 1329145223013136, and 1^{2} + 32^{2} + 9^{2} + 1^{2} + 4^{2} + 52^{2} + 2^{2} + 30^{2} + 1^{2} + 3^{2} + 1^{2} + 36^{2} = 6038.

by Yoshio Mimura, Kobe, Japan

## 6040

6040^{2} = 184^{3} + 228^{3} + 264^{3}.

by Yoshio Mimura, Kobe, Japan

## 6042

6042^{2} = 36505764, 3^{2} + 6^{2} + 5^{2} + 0^{2} + 5^{2} + 7^{2} + 6^{2} + 4^{2} = 14^{2}.

6042^{k} + 223554^{k} + 310156^{k} + 474297^{k} are squares for k = 1,2,3 (1007^{2}, 609235^{2}, 384324571^{2}).

6042^{5} = 8051997005161571232 : 8^{2} + 0^{2} + 5^{2} + 1^{2} + 9^{2} + 9^{2} + 70^{2} + 0^{2} + 5^{2} + 16^{2} + 1^{2} + 5^{2} + 7^{2} + 1^{2} + 23^{2} + 2^{2} = 6042.

by Yoshio Mimura, Kobe, Japan

## 6043

6043^{2} = 201^{3} + 236^{3} + 248^{3}.

6043^{2} = 36517849, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 6045

6045^{2} = 11^{3} + 253^{3} + 273^{3}.

by Yoshio Mimura, Kobe, Japan

## 6047

6047^{2} = 14^{4} + 14^{4} + 34^{4} + 77^{4}.

6047^{4} = 1337087640631681,

and 13^{2} + 3^{2} + 7^{2} + 0^{2} + 8^{2} + 7^{2} + 6^{2} + 40^{2} + 63^{2} + 1^{2} + 6^{2} + 8^{2} + 1^{2} = 6047.

by Yoshio Mimura, Kobe, Japan

## 6048

6048^{2} = (2^{2} - 1)(5^{2} - 1)(13^{2} - 1)(55^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(7^{2} - 1)(8^{2} - 1)(13^{2} - 1),br />
= (3^{2} - 1)(5^{2} - 1)(8^{2} - 1)(55^{2} - 1).

6048^{2} = 198^{3} + 234^{3} + 252^{3}.

6048^{4} = 1337972323516416,

and 13^{2} + 37^{2} + 9^{2} + 7^{2} + 2^{2} + 3^{2} + 2^{2} + 3^{2} + 51^{2} + 6^{2} + 41^{2} + 6^{2} = 6048.

by Yoshio Mimura, Kobe, Japan

## 6049

6049^{2} = 36590401, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 6050

6050^{2} = 55^{4} + 55^{4} + 55^{4} + 55^{4}.

by Yoshio Mimura, Kobe, Japan

## 6052

578^{k} + 2074^{k} + 2890^{k} + 4862^{k} are squares for k = 1,2,3 (102^{2}, 6052^{2}, 384948^{2}).

by Yoshio Mimura, Kobe, Japan

## 6056

Loop of length 35 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

6056 - 6736 - 5785 - 10474 - ... - 673 - 5365 - 7034 - 6056

(Note f(6056) = 60^{2} + 56^{2} = 6736, f(6736) = 67^{2} + 36^{2} = 5785, etc. See 37)

The square root of 6056 is 77.8203...., and 77 = 8^{2} + 2^{2} + 0^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 6057

6057^{2} = 14^{4} + 57^{4} + 58^{4} + 62^{4}.

6057^{2} = 36687249, 3 + 6 + 6 * 8 * 7 / 2 * 4 * 9 = 36 * 6 * 8 * 7 * 2 / 4 + 9 = 6057.

by Yoshio Mimura, Kobe, Japan

## 6058

6058^{2} = 36699364, 36 + 6 * 993 + 64 = 6058.

by Yoshio Mimura, Kobe, Japan

## 6059

6059^{4} = 1347732837213361, and 13^{2} + 47^{2} + 7^{2} + 3^{2} + 28^{2} + 37^{2} + 2^{2} + 13^{2} + 36^{2} + 1^{2} = 6059.

by Yoshio Mimura, Kobe, Japan

## 6060

Komachi equation: 6060^{2} = 98^{4} - 7^{4} - 6^{4} * 5^{4} - 43^{4} * 2^{4} + 1^{4}.

by Yoshio Mimura, Kobe, Japan

## 6061

6061^{2} = 32^{3} + 185^{3} + 312^{3}.

by Yoshio Mimura, Kobe, Japan

## 6062

1 / 6062 = 0.00016496205872...., 16^{2} + 4^{2} + 9^{2} + 6^{2} + 20^{2} + 5^{2} + 8^{2} + 72^{2} = 6062.

by Yoshio Mimura, Kobe, Japan

## 6063

6063^{2} = 8^{4} + 8^{4} + 51^{4} + 74^{4}.

Komachi equation: 6063^{2} = 987^{2} * 6^{2} * 5^{2} * 43^{2} / 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 6064

6064^{2} = (47 + 48 + 49 + 50)^{2} + (51 + 52 + 53 + 54)^{2} + (55 + 56 + 57 + 58)^{2} + ... + (299 + 300 + 301 + 302)^{2}.

by Yoshio Mimura, Kobe, Japan

## 6067

6067^{2} = 103^{3} + 251^{3} + 271^{3} = 153^{3} + 166^{3} + 306^{3}.

6067^{5} = 8219965120563755107 :

8^{2} + 2^{2} + 1^{2} + 9^{2} + 9^{2} + 6^{2} + 5^{2} + 1^{2} + 2^{2} + 0^{2} + 5^{2} + 6^{2} + 3^{2} + 75^{2} + 5^{2} + 1^{2} + 0^{2} + 7^{2} = 6067.

by Yoshio Mimura, Kobe, Japan

## 6068

6068^{2}± 3 are primes.

6068^{2} = 36820624, 3^{2} + 6^{2} + 8^{2} + 2^{2} + 0^{2} + 6^{2} + 2^{2} + 4^{2} = 13^{2}.

by Yoshio Mimura, Kobe, Japan

## 6069

6069^{2} = 161^{3} + 198^{3} + 292^{3}.

51^{k} + 1683^{k} + 6069^{k} + 10693^{k} are squares for k = 1,2,3 (136^{2}, 12410^{2}, 1204552^{2}).

by Yoshio Mimura, Kobe, Japan

## 6071

6071^{2} = 36857041, a square with different digita.

6071^{2} = 36857041, 368 + 5704 - 1 = 6071.

by Yoshio Mimura, Kobe, Japan

## 6072

6072^{2} = 74^{3} + 231^{3} + 289^{3} = 176^{3} + 196^{3} + 288^{3} = 191^{3} + 217^{3} + 270^{3}.

by Yoshio Mimura, Kobe, Japan

## 6075

6075^{2} = 105^{3} + 165^{3} + 315^{3} = 180^{3} + 225^{3} + 270^{3}.

by Yoshio Mimura, Kobe, Japan

## 6076

6076^{2} = (1^{2} + 3)(37^{2} + 3)(82^{2} + 3).

6076^{2} = 176^{3} + 238^{3} + 262^{3}.

by Yoshio Mimura, Kobe, Japan

## 6078

6078^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan

## 6080

Komchi equation: 6080^{2} = 9^{2} * 8^{2} * 76^{2} / 54^{2} * 3^{2} * 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 6081

6081^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 6082

6082^{2}± 3 are primes.

6082^{2} = 36990724 appears in the decimal expressions of π:

π = 3.14159•••36990724••• (from the 74009th digit)

(36990724 is the tenth 8-digit square in the expression of π.)

by Yoshio Mimura, Kobe, Japan

## 6083

6083^{2} = (29^{2} + 6)(209^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 6084

The square of 78.

6084^{2} = 102^{3} + 156^{3} + 318^{3}.

by Yoshio Mimura, Kobe, Japan

## 6085

6085^{2} = 124^{3} + 201^{3} + 300^{3}.

6085^{2} = 37027225, 3^{2} + 7^{2} + 0^{2} + 2^{2} + 7^{2} + 2^{2} + 2^{2} + 5^{2} = 12^{2}.

6085^{2} = 37027225, 3 + 7 + 0 + 27 * 225 = 6085.

by Yoshio Mimura, Kobe, Japan

## 6086

6086^{2} = S_{2}(56) + S_{2}(480), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 6087

6087^{2} = 16^{4} + 22^{4} + 51^{4} + 74^{4}.

by Yoshio Mimura, Kobe, Japan

## 6088

6088^{5} = 8363214174283399168 : 8^{2} + 3^{2} + 63^{2} + 21^{2} + 4^{2} + 17^{2} + 4^{2} + 28^{2} + 3^{2} + 3^{2} + 9^{2} + 9^{2} + 16^{2} + 8^{2} = 8^{2} + 3^{2} + 63^{2} + 2^{2} + 14^{2} + 1^{2} + 7^{2} + 4^{2} + 2^{2} + 8^{2} + 3^{2} + 39^{2} + 9^{2} + 1^{2} + 6^{2} + 8^{2} = 6088.

by Yoshio Mimura, Kobe, Japan

## 6089

1 / 6089 = 0.00016423057973...., 1^{2} + 6^{2} + 42^{2} + 30^{2} + 57^{2} + 9^{2} + 7^{2} + 3^{2} = 6089.

by Yoshio Mimura, Kobe, Japan

## 6090

6090^{2} = (1^{2} + 6)(2^{2} + 6)(3^{2} + 6)(9^{2} + 6)(20^{2} + 6) = (1^{2} + 6)(2^{2} + 6)(9^{2} + 6)(78^{2} + 6)

= (1^{2} + 6)(6^{2} + 6)(9^{2} + 6)(38^{2} + 6) = (1^{2} + 6)(9^{2} + 6)(12^{2} + 6)(20^{2} + 6)

= (10^{2} + 5)(13^{2} + 5)(45^{2} + 5) = (3^{2} + 6)(20^{2} + 6)(78^{2} + 6) = (3^{2} + 6)(8^{2} + 6)(9^{2} + 6)(20^{2} + 6)

= (8^{2} + 6)(9^{2} + 6)(78^{2} + 6).

The integral triangle of sides 10585, 355251, 365834 has square area 6090^{2}.

Komachi equation: 6090^{2} = 9^{2} * 87^{2} * 6^{2} / 54^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 6092

6092^{4} = 1377334984151296,

and 1^{2} + 37^{2} + 7^{2} + 33^{2} + 49^{2} + 8^{2} + 4^{2} + 15^{2} + 1^{2} + 29^{2} + 6^{2} = 6092.

6092^{5} = 8390724723449695232 :

8^{2} + 3^{2} + 9^{2} + 0^{2} + 7^{2} + 24^{2} + 7^{2} + 2^{2} + 3^{2} + 4^{2} + 49^{2} + 6^{2} + 9^{2} + 52^{2} + 3^{2} + 2^{2} = 6092.

by Yoshio Mimura, Kobe, Japan

## 6095

6095^{2} = 37149025, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 6096

6096^{2} = 37161216, a zigzag square.

6096 = (1^{2} + 2^{2} + 3^{2} + ... + 63^{2}) / (1^{2} + 2^{2} + 3^{2}).

by Yoshio Mimura, Kobe, Japan

## 6097

6097^{2} = 37173409, a zigzag square.

1716^{k} + 2184^{k} + 6097^{k} + 10452^{k} are squares for k = 1,2,3 (143^{2}, 12415^{2}, 1176409^{2}).

by Yoshio Mimura, Kobe, Japan

## 6098

6098^{2} = 37185604, a zigzag square with different digits.

by Yoshio Mimura, Kobe, Japan

## 6099

6099^{2} = 37197801, a zigzag square.

60996100 = 7810^{2}.

by Yoshio Mimura, Kobe, Japan