## 7100

7100^{2} = 6^{3} + 55^{3} + 369^{3} = 94^{3} + 278^{3} + 304^{3}.

by Yoshio Mimura, Kobe, Japan

## 7102

7102^{2} = 60^{3} + 147^{3} + 361^{3}.

by Yoshio Mimura, Kobe, Japan

## 7106

7106^{2} = 50495236, and 5^{2} + 0^{2} + 4^{2} + 9^{2} + 5^{2} + 2^{2} + 3^{2} + 6^{2} = 14^{2}.

7106^{k} + 33022^{k} + 40546^{k} + 94050^{k} are squares for k = 1,2,3 (418^{2}, 107844^{2}, 30576700^{2}).

638^{k} + 4378^{k} + 7106^{k} + 11594^{k} are squares for k = 1,2,3 (154^{2}, 14300^{2}, 1414732^{2}).

by Yoshio Mimura, Kobe, Japan

## 7108

7108^{2} = 32^{4} + 58^{4} + 64^{4} + 68^{4}.

by Yoshio Mimura, Kobe, Japan

## 7109

7109^{5} = 18156936350131874549 :

18^{2} + 1^{2} + 5^{2} + 6^{2} + 9^{2} + 3^{2} + 6^{2} + 3^{2} + 50^{2} + 1^{2} + 31^{2} + 8^{2} + 7^{2} + 4^{2} + 54^{2} + 9^{2} = 7109.

by Yoshio Mimura, Kobe, Japan

## 7111

7111^{5} = 18182491499287016551 :

1^{2} + 81^{2} + 8^{2} + 2^{2} + 4^{2} + 9^{2} + 1^{2} + 4^{2} + 9^{2} + 9^{2} + 2^{2} + 8^{2} + 7^{2} + 0^{2} + 1^{2} + 6^{2} + 5^{2} + 5^{2} + 1^{2} = 7111.

by Yoshio Mimura, Kobe, Japan

## 7112

7112^{2} = (1^{2} + 7)(7^{2} + 7)(336^{2} + 7) = (21^{2} + 7)(336^{2} + 7).

by Yoshio Mimura, Kobe, Japan

## 7113

7113^{5} = 18208075414611986793 :

1^{2} + 8^{2} + 2^{2} + 0^{2} + 80^{2} + 7^{2} + 5^{2} + 4^{2} + 14^{2} + 6^{2} + 1^{2} + 1^{2} + 9^{2} + 8^{2} + 6^{2} + 7^{2} + 9^{2} + 3^{2} = 7113.

by Yoshio Mimura, Kobe, Japan

## 7114

(7114^{2} + 4) = (1^{2} + 4)(3^{2} + 4)(6^{2} + 4)(9^{2} + 4)(15^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 7115

7115^{2} = 50^{3} + 246^{3} + 329^{3}.

by Yoshio Mimura, Kobe, Japan

## 7116

7116^{2} = 50637456, and 5^{2} + 0^{2} + 6^{2} + 3^{2} + 7^{2} + 4^{2} + 5^{2} + 6^{2} = 14^{2}.

7116^{2} = 4^{3} + 198^{3} + 350^{3} = 49^{3} + 162^{3} + 359^{3}.

by Yoshio Mimura, Kobe, Japan

## 7118

7118^{5} = 18272161214279737568 :

18^{2} + 27^{2} + 2^{2} + 1^{2} + 6^{2} + 1^{2} + 21^{2} + 4^{2} + 27^{2} + 9^{2} + 7^{2} + 3^{2} + 7^{2} + 5^{2} + 68^{2}

= 18^{2} + 27^{2} + 21^{2} + 6^{2} + 1^{2} + 2^{2} + 1^{2} + 4^{2} + 27^{2} + 9^{2} + 7^{2} + 3^{2} + 7^{2} + 5^{2} + 68^{2} = 7118.

Komachi equations: 7118^{2} = 358^{3} + 3^{5} + 9^{7}.

by Yoshio Mimura, Kobe, Japan

## 7121

7121^{3} = 361096232561, and 3^{2} + 6^{2} + 1^{2} + 0^{2} + 9^{2} + 62^{2} + 3^{2} + 2^{2} + 56^{2} + 1^{2} = 7121.

Komachi equation: 7120^{2} = 9^{4} * 8^{4} + 76^{4} - 54^{4} - 32^{4} + 10^{4}.

by Yoshio Mimura, Kobe, Japan

## 7122

7122^{5} = 18323559666218041632 :

1^{2} + 8^{2} + 32^{2} + 3^{2} + 55^{2} + 9^{2} + 6^{2} + 6^{2} + 6^{2} + 2^{2} + 1^{2} + 8^{2} + 0^{2} + 41^{2} + 6^{2} + 32^{2} = 7122.

by Yoshio Mimura, Kobe, Japan

## 7124

Loop of length 5 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

7124 - 5617 - 3425 - 1781 - ... - 3425 - 1781 - 6850 - 7124

(Note f(7124) = 71^{2} + 24^{2} = 5617, f(5617) = 56^{2} + 17^{2} = 3425, etc. See 1781)

by Yoshio Mimura, Kobe, Japan

## 7127

7127^{2} = 72^{3} + 150^{3} + 361^{3} = 157^{3} + 247^{3} + 317^{3}.

by Yoshio Mimura, Kobe, Japan

## 7128

7128^{2} = 50808384, a zigzag square.

7128^{2} = 4! + 5! + 7! + 10! + 10! + 10! + 11!

7128^{2} = (1^{2} + 8)(4^{2} + 8)(17^{2} + 8)(28^{2} + 8) = (1^{2} + 8)(8^{2} + 8)(16^{2} + 8)(17^{2} + 8)

= (1^{2} + 8)(8^{2} + 8)(280^{2} + 8) = (5^{2} + 8)(28^{2} + 8)(44^{2} + 8).

7128^{2} = 90^{3} + 216^{3} + 342^{3}.

by Yoshio Mimura, Kobe, Japan

## 7130

7130^{5} = 18426703563479300000 :

1^{2} + 8^{2} + 4^{2} + 2^{2} + 67^{2} + 0^{2} + 35^{2} + 6^{2} + 34^{2} + 7^{2} + 9^{2} + 3^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 7130.

by Yoshio Mimura, Kobe, Japan

## 7132

7132^{5} = 18452561970246802432 :

18^{2} + 4^{2} + 5^{2} + 2^{2} + 5^{2} + 6^{2} + 1^{2} + 9^{2} + 70^{2} + 2^{2} + 4^{2} + 6^{2} + 8^{2} + 0^{2} + 24^{2} + 32^{2}

= 18^{2} + 4^{2} + 5^{2} + 2^{2} + 5^{2} + 6^{2} + 1^{2} + 9^{2} + 70^{2} + 24^{2} + 6^{2} + 8^{2} + 0^{2} + 2^{2} + 4^{2} + 32^{2} = 7132.

by Yoshio Mimura, Kobe, Japan

## 7133

7133^{2} = 50879689, and 5^{2} + 0^{2} + 8^{2} + 7^{2} + 9^{2} + 6^{2} + 8^{2} + 9^{2} = 20^{2}.

by Yoshio Mimura, Kobe, Japan

## 7134

2958^{k} + 7134^{k} + 24708^{k} + 33321^{k} are squares for k = 1,2,3 (261^{2}, 42195^{2}, 7243533^{2}).

by Yoshio Mimura, Kobe, Japan

## 7136

7136^{2}± 3 are primes.

7136^{5} = 18504365873536434176 :

1^{2} + 8^{2} + 5^{2} + 0^{2} + 4^{2} + 3^{2} + 6^{2} + 5^{2} + 8^{2} + 73^{2} + 5^{2} + 3^{2} + 6^{2} + 4^{2} + 34^{2} + 17^{2} + 6^{2}

= 1^{2} + 8^{2} + 50^{2} + 4^{2} + 36^{2} + 5^{2} + 8^{2} + 7^{2} + 3^{2} + 5^{2} + 36^{2} + 4^{2} + 3^{2} + 41^{2} + 7^{2} + 6^{2} = 7136.

by Yoshio Mimura, Kobe, Japan

## 7140

7140^{2} = (9^{2} - 1)(16^{2} - 1)(50^{2} - 1).

7140^{2} = 198 * 199 * 200 + 200 * 201 * 202 + 202 * 203 * 204 + ... + 208 * 209 * 210..

The integral triangle of sides 7208, 14295, 16913 (or 6713, 15912, 19025)(or 1433, 77600, 78183) has square area 7140^{2}.

Komachi equation: 7140^{2} = 1^{2} / 2^{2} * 34^{2} * 5^{2} / 6^{2} * 7^{2} * 8^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 7141

Loop of length 6 by the function f(N) = b^{2} + a^{2} for N = 100b + a:

7141 - 6722 - 4973 - 7730 - 6829 - 5465 - 7141 (Note f(7141) = 71^{2} + 41^{2} = 6722.)

by Yoshio Mimura, Kobe, Japan

## 7144

7144^{2} = 51036736, and 51^{2} + 0^{2} + 3^{2} + 67^{2} + 3^{2} + 6^{2}.

7144 = (1^{2} + 2^{2} + 3^{2} + ... + 47^{2}) / (1^{2} + 2^{2}).

7144^{5} = 18608322723436724224 :

18^{2} + 6^{2} + 0^{2} + 8^{2} + 32^{2} + 2^{2} + 7^{2} + 23^{2} + 4^{2} + 3^{2} + 67^{2} + 2^{2} + 4^{2} + 2^{2} + 24^{2}

= 18^{2} + 6^{2} + 0^{2} + 8^{2} + 3^{2} + 22^{2} + 7^{2} + 2^{2} + 34^{2} + 3^{2} + 67^{2} + 2^{2} + 4^{2} + 22^{2} + 4^{2}

= 18^{2} + 6^{2} + 0^{2} + 8^{2} + 32^{2} + 2^{2} + 7^{2} + 23^{2} + 4^{2} + 3^{2} + 67^{2} + 24^{2} + 2^{2} + 2^{2} + 4^{2} = 7144.

by Yoshio Mimura, Kobe, Japan

## 7145

7145^{2} = 51051025, and 5^{2} + 1^{2} + 0^{2} + 5^{2} + 1^{2} + 0^{2} + 2^{2} + 5^{2} = 9^{2}.

7145^{2} = 75^{3} + 201^{3} + 349^{3}.

by Yoshio Mimura, Kobe, Japan

## 7152

7152^{2} = 20^{3} + 208^{3} + 348^{3}.

by Yoshio Mimura, Kobe, Japan

## 7154

7154^{2} = 51179716, a square with odd digits except the last digit 6.

by Yoshio Mimura, Kobe, Japan

## 7155

7155^{2} = (260 + 261 + 262)^{2} + (263 + 264 + 265)^{2} + (266 + 267 + 268)^{2} + ... + (407 + 408 + 409)^{2}.

7155^{4} = 2620828195700625,

and 26^{2} + 20^{2} + 8^{2} + 2^{2} + 8^{2} + 19^{2} + 5^{2} + 70^{2} + 0^{2} + 6^{2} + 25^{2} = 7155.

by Yoshio Mimura, Kobe, Japan

## 7161

496^{k} + 5270^{k} + 7161^{k} + 11098^{k} are squares for k = 1,2,3 (155^{2}, 14229^{2}, 1371347^{2}).

by Yoshio Mimura, Kobe, Japan

## 7166

7166^{2} = 51351556, a square with odd digits except the last digit 6.

7166^{2} = 5^{5} + 15^{5} + 22^{5} + 34^{5}.

by Yoshio Mimura, Kobe, Japan

## 7167

7167^{5} = 18909803779633220607 :

18^{2} + 9^{2} + 0^{2} + 9^{2} + 8^{2} + 0^{2} + 37^{2} + 7^{2} + 9^{2} + 6^{2} + 3^{2} + 32^{2} + 20^{2} + 60^{2} + 7^{2} = 7167.

by Yoshio Mimura, Kobe, Japan

## 7168

7168^{2} = (3^{2} + 7)(7^{2} + 7)(11^{2} + 7)(21^{2} + 7).

7168^{2} = 32^{4} + 64^{4} + 64^{4} + 64^{4}.

by Yoshio Mimura, Kobe, Japan

## 7171

7171^{2} = 51423241, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 7172

7172^{2} = 51437584, a zigzag square.

7172^{2} = 8^{4} + 16^{4} + 68^{4} + 74^{4}.

7172^{3} = 368910352448, and 3^{2} + 68^{2} + 9^{2} + 10^{2} + 3^{2} + 5^{2} + 2^{2} + 4^{2} + 48^{2} = 7172.

by Yoshio Mimura, Kobe, Japan

## 7173

7173^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 7176

7176^{2} = 23^{3} + 169^{3} + 360^{3} = 100^{3} + 272^{3} + 312^{3} = 122^{3} + 156^{3} + 358^{3}.

by Yoshio Mimura, Kobe, Japan

## 7177

7177^{2} = 230^{3} + 236^{3} + 297^{3}.

by Yoshio Mimura, Kobe, Japan

## 7179

7179^{2} = 51538041, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 7180

7180^{2} = 185 x 186 + 186 x 187 + 187 x 188 + ... + 543 x 544.

by Yoshio Mimura, Kobe, Japan

## 7181

7181^{2} = 64^{3} + 102^{3} + 369^{3}.

by Yoshio Mimura, Kobe, Japan

## 7182

7182^{2} = 17^{4} + 43^{4} + 69^{4} + 71^{4}.

by Yoshio Mimura, Kobe, Japan

## 7184

7184^{4} = 2663577236340736,

and 2^{2} + 6^{2} + 6^{2} + 3^{2} + 57^{2} + 7^{2} + 2^{2} + 36^{2} + 34^{2} + 0^{2} + 7^{2} + 36^{2} = 7184.

by Yoshio Mimura, Kobe, Japan

## 7186

7186^{2} = 51638596, a zigzag square.

7186^{2} = 25^{4} + 45^{4} + 45^{4} + 81^{4}.

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

7186 - 12437 - 1946 - 2477 - ... - 8731 - 8530 - 8125 - 7186

(Note f(7186) = 71^{2} + 86^{2} = 12437, f(12437) = 1^{2} + 24^{2} + 37^{2} = 1946, etc. See 1946)

by Yoshio Mimura, Kobe, Japan

## 7187

7187^{2} = 51652969, and 5^{2} + 1^{2} + 6^{2} + 5^{2} + 2^{2} + 9^{2} + 6^{2} + 9^{2} = 17^{2}.

by Yoshio Mimura, Kobe, Japan

## 7188

1 / 7188 = 0.000139120756, 1^{2} + 39^{2} + 1^{2} + 2^{2} + 0^{2} + 75^{2} + 6^{2} = 7188,

1 / 7188 = 0.000139120756, 1^{2} + 39^{2} + 1^{2} + 2^{2} + 075^{2} + 6^{2} = 7188.

7188^{2} = 16^{3} + 194^{3} + 354^{3}.

7188^{3} = 371384868672, and 3^{2} + 7^{2} + 13^{2} + 8^{2} + 48^{2} + 6^{2} + 8^{2} + 67^{2} + 2^{2} = 7188.

by Yoshio Mimura, Kobe, Japan

## 7190

7190^{2} = 99^{3} + 249^{3} + 328^{3}.

by Yoshio Mimura, Kobe, Japan

## 7191

7191^{2} = 51710481, and 5 + 1 + 7104 + 81 = 7191.

by Yoshio Mimura, Kobe, Japan

## 7192

7192^{2} = 196^{3} + 240^{3} + 312^{3}.

by Yoshio Mimura, Kobe, Japan

## 7196

7196^{2} = 51782416, and 5^{2} + 1^{2} + 7^{2} + 8^{2} + 2^{2} + 4^{2} + 1^{2} + 6^{2} = 14^{2}.

7196^{2} = 194^{3} + 207^{3} + 329^{3}.

by Yoshio Mimura, Kobe, Japan

## 7197

7197^{2} = 44^{3} + 220^{3} + 345^{3}.

by Yoshio Mimura, Kobe, Japan

## 7198

7198^{k} + 16874^{k} + 20650^{k} + 80594^{k} are squares for k = 1,2,3 (354^{2}, 85196^{2}, 23183460^{2}).

by Yoshio Mimura, Kobe, Japan