## 2200

2200 = (1^{2} + 2^{2} + 3^{2} + ... + 175^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 13^{2}).

462^{k} + 1034^{k} + 2200^{k} + 2233^{k} are squares for k = 1,2,3 (77^{2}, 3333^{2}, 151613^{2}).

by Yoshio Mimura, Kobe, Japan

## 2201

2201^{2} = 4844401, a reversible square (1044484 = 1022^{2}).

by Yoshio Mimura, Kobe, Japan

## 2202

2202^{2} = 4848804, a square with just 3 kinds of even digits.

2202^{2} = 4848804, a reversible square (4088484 = 2022^{2}).

by Yoshio Mimura, Kobe, Japan

## 2203

2203^{2} = 4853209, a squaare with different digits.

by Yoshio Mimura, Kobe, Japan

## 2205

2205^{2} = 7^{3} + 119^{3} + 147^{3} = 14^{3} + 49^{3} + 168^{3} = 49^{3} + 126^{3} + 140^{3}.

A cubic polynomial :

(X + 32^{2})(X + 2205^{2})(X + 4224^{2}) = X^{3} + 4765^{2}X^{2} + 9315168^{2}X + 298045440^{2}.

2205^{2} = (20 + 21 + 22)^{2} + (23 + 24 + 25)^{2} + (26 + 27 + 28)^{2} + (29 + 30 + 31)^{2} + ... + (167 + 168 + 169)^{2}.

2205^{2} = (53 + 54 + 55 + ... + 73)^{2} + (74 + 75 + 76 + ... + 94)^{2} + (95 + 96 + 97 + ... + 115)^{2} + ... + (74 + 75 + 76 + ... + 94)^{2}.

2205^{2} = 287^{2} + 288^{2} + 289^{2} + 290^{2} + 291^{2} + 292^{2} + 293^{2} + ... + 336^{2}.

by Yoshio Mimura, Kobe, Japan

## 2206

2206^{2} = 19^{3} + 32^{3} + 169^{3}.

by Yoshio Mimura, Kobe, Japan

## 2209

The square of 47.

2209^{2} = 4879681, a zigzag square.

Komachi equation: 2209^{2} = 987^{4} / 6^{4} * 5^{4} * 4^{4} * 3^{4} / 210^{4}.

2209^{2} = 4879681 appears in the decimal expressions of e:

e = 2.71828•••4879681••• (from the 10075th digit)

(4879681 is the fifth 7-digit square in the expression of e.)

by Yoshio Mimura, Kobe, Japan

## 2210

2210 = (1^{2} + 2^{2} + 3^{2} + ... + 84^{2}) / (1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2}).

2210^{2} = (1^{2} + 1)(2^{2} + 1)(4^{2} + 1)(8^{2} + 1)(21^{2} + 1)

= (1^{2} + 1)(4^{2} + 1)(18^{2} + 1)(21^{2} + 1) = (1^{2} + 1)(4^{2} + 1)(8^{2} + 1)(47^{2} + 1)

= (1^{2} + 1)(5^{2} + 1)(8^{2} + 1)(38^{2} + 1) = (2^{2} + 1)(21^{2} + 1)(47^{2} + 1)

= (2^{2} + 1)(4^{2} + 1)(5^{2} + 1)(47^{2} + 1) = (3^{2} + 1)(4^{2} + 1)(8^{2} + 1)(21^{2} + 1)

= (4^{2} + 1)(5^{2} + 1)(8^{2} + 1)(13^{2} + 1) = (8^{2} + 1)(13^{2} + 1)(21^{2} + 1)

= (8^{2} + 4)(9^{2} + 4)(29^{2} + 4) = (29^{2} + 4)(76^{2} + 4) = (5^{2} + 9)(379^{2} + 9).

1666^{k} + 2210^{k} + 5270^{k} + 9350^{k} are squares for k = 1,2,3 (136^{2}, 11084^{2}, 989536^{2}).

by Yoshio Mimura, Kobe, Japan

## 2211

2211^{2} = 4888521, a reversible square (1258884 = 1122^{2}).

2211^{2}± 2 are primes.

2211^{2} = 92^{3} + 102^{3} + 145^{3}.

2211^{2} + 2212^{2} + 2213^{2} + ... + 2244^{2} = 2245^{2} + 2246^{2} + 2247^{2} + ... + 2277^{2}.

by Yoshio Mimura, Kobe, Japan

## 2214

2214^{5} = 53197115312713824 : 5^{2} + 3^{2} + 1^{2} + 9^{2} + 7^{2} + 1^{2} + 15^{2} + 31^{2} + 2^{2} + 7^{2} + 13^{2} + 8^{2} + 24^{2} = 5^{2} + 31^{2} + 9^{2} + 7^{2} + 1^{2} + 15^{2} + 3^{2} + 1^{2} + 2^{2} + 7^{2} + 13^{2} + 8^{2} + 24^{2} = 2214.

2214^{5} = (1^{2} + 5)(2^{2} + 5)(6^{2} + 5)(47^{2} + 5) = (6^{2} + 5)(7^{2} + 5)(47^{2} + 5).

by Yoshio Mimura, Kobe, Japan

## 2216

2216^{2} = 4910656, and 4 * 9 * 10 * 6 + 56 = 2216.

by Yoshio Mimura, Kobe, Japan

## 2218

2218^{2} = 17^{3} + 99^{3} + 158^{3}.

by Yoshio Mimura, Kobe, Japan

## 2219

2219^{4} = 24245391929521, and 2^{2} + 4^{2} + 2^{2} + 4^{2} + 5^{2} + 39^{2} + 1^{2} + 9^{2} + 2^{2} + 9^{2} + 5^{2} + 21^{2} = 2219.

by Yoshio Mimura, Kobe, Japan

## 2222

2222^{2} = 4937284, a zigzag square.

2222^{2}± 3 are primes.

2222^{2} = 192^{2} + 193^{2} + 194^{2} + 195^{2} + 196^{2} + 197^{2} + ... + 279^{2}.

2222^{2} = 58^{3} + 63^{3} + 165^{3}.

by Yoshio Mimura, Kobe, Japan

## 2223

2223^{2} = 4941729, and 494 + 1729 = 2223.

2223^{2} = 9^{3} + 130^{3} + 140^{3}.

by Yoshio Mimura, Kobe, Japan

## 2224

2224^{2} = 4946176, a zigzag square.

2224^{2} = 24^{3} + 97^{3} + 159^{3} = 87^{3} + 97^{3} + 150^{3}.

by Yoshio Mimura, Kobe, Japan

## 2225

2225^{2} = (11^{2} + 4)(199^{2} + 4).

2225^{2} = 4950625, and 4 = 2^{2}, 950625 = 975^{2},

2225^{2} = 4950625, and 49 = 7^{2}, 50625 = 225^{2}.

by Yoshio Mimura, Kobe, Japan

## 2227

2227^{2} = S_{2}(43) + S_{2}(245), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 2228

2228^{2} = 75^{3} + 101^{3} + 152^{3}.

1 / 2228 = 0.00044883303411, 44^{2} + 8^{2} + 8^{2} + 3^{2} + 3^{2} + 03^{2} + 4^{2} + 11^{2} = 2228.

Loop of length 10 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

2228 - 1268 - 4768 - 6833 - ... - 6596 - 13441 - 2838 - 2228

(Note f(2228) = 22^{2} + 28^{2} = 1268, f(1268) = 12^{2} + 68^{2} = 4768, etc. See 1268)

by Yoshio Mimura, Kobe, Japan

## 2230

2230^{2} = 4972900, and 49 = 7^{2}, 72900 = 270^{2}.

by Yoshio Mimura, Kobe, Japan

## 2232

2232^{2} = 84^{3} + 84^{3} + 156^{3}.

by Yoshio Mimura, Kobe, Japan

## 2233

462^{k} + 1034^{k} + 2200^{k} + 2233^{k} are squares for k = 1,2,3 (77^{2}, 3333^{2}, 151613^{2}).

by Yoshio Mimura, Kobe, Japan

## 2234

2234^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 2236

2236^{2} = 4999696, a square with just 3 kinds of digits.

2236^{2} = 4999696, a square pegged by 9.

by Yoshio Mimura, Kobe, Japan

## 2237

2237^{2} = 113^{3} + 116^{3} + 126^{3}.

by Yoshio Mimura, Kobe, Japan

## 2239

2239^{3} = 11224377919, and 11^{2} + 2^{2} + 2^{2} + 43^{2} + 7^{2} + 7^{2} + 9^{2} + 1^{2} + 9^{2} = 2239.

by Yoshio Mimura, Kobe, Japan

## 2240

2240^{2} = 24^{3} + 64^{3} + 168^{3} = 2^{7} + 2^{7} + 5^{7} + 5^{7} + 5^{7} + 9^{7}.

2240^{2} = (3^{2} - 1)(6^{2} - 1)(9^{2} - 1)(15^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 2241

2241^{2} = 106^{3} + 108^{3} + 137^{3}.

by Yoshio Mimura, Kobe, Japan

## 2242

2242^{2} = 3^{0} + 3^{3} + 3^{4} + 3^{6} + 3^{8} + 3^{10} + 3^{11} + 3^{14}.

2242^{k} + 4978^{k} + 13262^{k} + 15618^{k} are squares for k = 1,2,3 (190^{2}, 21204^{2}, 2505340^{2}).

by Yoshio Mimura, Kobe, Japan

## 2243

2243^{5} = 56773591412619443 : 5^{2} + 6^{2} + 7^{2} + 7^{2} + 35^{2} + 9^{2} + 14^{2} + 12^{2} + 6^{2} + 19^{2} + 4^{2} + 4^{2} + 3^{2} = 2243.

by Yoshio Mimura, Kobe, Japan

## 2244

2244^{2} = (3^{2} + 8)(14^{2} + 8)(38^{2} + 8) = (3^{2} + 8)(5^{2} + 8)(6^{2} + 8)(14^{2} + 8) = (3^{2} + 8)(6^{2} + 8)(82^{2} + 8).

2244^{2} = S_{2}(50) + S_{2}(246), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 2245

2245^{5} = 57027157741403125 : 5^{2} + 7^{2} + 0^{2} + 2^{2} + 7^{2} + 15^{2} + 7^{2} + 7^{2} + 4^{2} + 1^{2} + 40^{2} + 3^{2} + 12^{2} + 5^{2} = 2245.

by Yoshio Mimura, Kobe, Japan

## 2247

2247^{2} = 2^{3} + 126^{3} + 145^{3}.

by Yoshio Mimura, Kobe, Japan

## 2248

2248^{2} = 5053504, a zigzag square.

2248^{2}± 3 are primes.

2248^{2} = 5053504, 5^{2} + 0^{2} + 5^{2} + 3^{2} + 5^{2} + 0^{2} + 4^{2} = 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 2250

2250^{2} = 57^{3} + 107^{3} + 154^{3}.

2250^{2} = (1^{2} + 9)(4^{2} + 9)(6^{2} + 9)(21^{2} + 9) = (3^{2} + 9)(6^{2} + 9)(79^{2} + 9).

by Yoshio Mimura, Kobe, Japan

## 2251

2251^{2} = 1^{5} + 9^{5} + 15^{5} + 16^{5} + 20^{5}.

by Yoshio Mimura, Kobe, Japan

## 2252

2252^{2} = 5071504, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 2254

2254^{2} = 5080516, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 2256

2256^{2} = 46 x 47 x 48 + 48 x 49 x 50 + 50 x 51 x 52 + 52 x 53 x 54 + ... + 80 x 81 x 82.

by Yoshio Mimura, Kobe, Japan

## 2257

2257^{2} = 4^{4} + 31^{4} + 38^{4} + 38^{4}.

2257^{2} = 225^{2} + 226^{2} + 227^{2} + 228^{2} + 229^{2} + 230^{2} + ... + 298^{2}.

1517^{k} + 2257^{k} + 12173^{k} + 33337^{k} are squares for k = 1,2,3 (222^{2}, 35594^{2}, 6234426^{2}).

1 / 2257 = 0.0004430660168, 4^{2} + 43^{2} + 0^{2} + 6^{2} + 6^{2} + 0^{2} + 16^{2} + 8^{2} = 2257,

1 / 2257 = 0.0004430660168, 4^{2} + 43^{2} + 0^{2} + 6^{2} + 6^{2} + 016^{2} + 8^{2} = 2257,

1 / 2257 = 0.0004430660168, 4^{2} + 43^{2} + 06^{2} + 6^{2} + 0^{2} + 16^{2} + 8^{2} = 2257,

1 / 2257 = 0.0004430660168, 4^{2} + 43^{2} + 06^{2} + 6^{2} + 016^{2} + 8^{2} = 2257.

by Yoshio Mimura, Kobe, Japan

## 2259

2259^{2} = 5103081, 5^{2} + 1^{2} + 0^{2} + 3^{2} + 0^{2} + 8^{2} + 1^{2} = 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 2260

2260^{2} = 67^{3} + 109^{3} + 152^{3}.

Komachi equations:

2260^{2} = 1^{2} / 2^{2} * 3^{2} * 4^{2} * 5^{2} * 678^{2} / 9^{2} = 1^{2} * 2^{2} / 3^{2} * 45^{2} * 678^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 2261

2261^{2} = 5112121, a square with just 3 kinds of digits.

2261^{2} = 166^{3} + 14^{5} + 1^{7}.

22^{k} + 73^{k} + 130^{k} + 136^{k} are squares for k = 1,2,3 (19^{2}, 203^{2}, 2261^{2}).

by Yoshio Mimura, Kobe, Japan

## 2262

2262^{2} = (7^{2} + 9)(297^{2} + 9).

390^{k} + 2262^{k} + 4329^{k} + 6708^{k} are squares for k = 1,2,3 (117^{2}, 8307^{2}, 628173^{2}).

by Yoshio Mimura, Kobe, Japan

## 2263

1 / 2263 = 0.000441...., and 441 = 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 2264

2264^{2}± 3 are primes.

2264^{2} = 84^{3} + 107^{3} + 149^{3}.

1 / 2264 = 0.000441...., and 441 = 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 2265

1 / 2265 = 0.000441...., and 441 = 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 2267

1 / 2267 = 0.000441...., and 441 = 21^{2}.

by Yoshio Mimura, Kobe, Japan

## 2268

2268^{2} = 5143824, a zigzag square.

2268^{2} = 54^{3} + 126^{3} + 144^{3} = 81^{3} + 84^{3} + 159^{3} = 12^{4} + 24^{4} + 36^{4} + 42^{4} = 18^{4} + 36^{4} + 36^{4} + 36^{4}.

2268^{2} = (1^{2} + 5)(2^{2} + 5)(3^{2} + 5)(7^{2} + 5)(11^{2} + 5).

A cubic polynomial :

(X + 896^{2})(X + 1269^{2})(X + 2268^{2}) = X^{3} + 2749^{2}X^{2} + 3702132^{2}X + 2578770432^{2}.

Komachi equations:

2268^{2} = 9^{2} / 8^{2} * 7^{2} / 6^{2} * 54^{2} * 32^{2} */ 1^{2} = 9^{2} * 8^{2} * 7^{2} * 6^{2} * 5^{2} / 4^{2} * 3^{2} * 2^{2} / 10^{2}

= 9^{2} * 8^{2} * 7^{2} * 6^{2} / 5^{2} / 4^{2} * 3^{2} / 2^{2} * 10^{2} = 9^{2} / 8^{2} * 7^{2} * 6^{2} / 5^{2} * 4^{2} * 3^{2} * 2^{2} * 10^{2}

= - 9^{2} * 8^{2} * 7^{2} * 6^{2} + 54^{2} / 3^{2} * 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 2274

2274^{2} = 5^{3} + 120^{3} + 151^{3} = 7^{5} + 11^{5} + 13^{5} + 14^{5} + 21^{5}.

by Yoshio Mimura, Kobe, Japan

## 2275

2275^{2} = 5175625, a zigzag square.

2275^{2} = 5175625, and 517 * 5 - 62 * 5 = 5 * 1 * 75 * 6 + 25 = 5 / 1 * 75 * 6 + 25 = 2275.

2275^{2} = (8 + 9 + 10 + ... + 42)^{2} + (43 + 44 + 45 + ... + 77)^{2} + (78 + 79 + 80 + ... + 112)^{2} + ... + (43 + 44 + 45 + ... + 77)^{2}.

by Yoshio Mimura, Kobe, Japan

## 2277

2277^{2} = 5184729, a zigzag square with different digits.

2277^{2} = 5184729, and 5184 = 72^{2}, 729 = 27^{2}.

1 / 2277 = 0.0004391743522, 4^{2} + 3^{2} + 9^{2} + 17^{2} + 43^{2} + 5^{2} + 2^{2} + 2^{2} = 2277,

1 / 2277 = 0.0004391743522, 43^{2} + 9^{2} + 17^{2} + 4^{2} + 3^{2} + 5^{2} + 2^{2} + 2^{2} = 2277.

2277^{2} = (80 + 81 + 82)^{2} + (83 + 84 + 85)^{2} + (86 + 87 + 88)^{2} + (89 + 90 + 91)^{2} + ... + (176 + 177 + 178)^{2}.

by Yoshio Mimura, Kobe, Japan

## 2280

2280^{2} = 70 x 71 x 72 + 72 x 73 x 74 + 74 x 75 x 76 + 76 x 77 x 78 + ... + 88 x 89 x 90.

2280^{2} = 10^{3} + 27^{3} + 173^{3} = 19^{3} + 24^{3} + 173^{3} = 38^{3} + 136^{3} + 138^{3}.

by Yoshio Mimura, Kobe, Japan

## 2282

2282^{4} = 27118306210576, and 27^{2} + 1^{2} + 1^{2} + 8^{2} + 30^{2} + 6^{2} + 21^{2} + 0^{2} + 5^{2} + 7^{2} + 6^{2} = 2282.

2282^{k} + 3010^{k} + 3626^{k} + 6958^{k} are squares for k = 1,2,3 (126^{2}, 8708^{2}, 650916^{2}).

by Yoshio Mimura, Kobe, Japan

## 2283

2283^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 2285

2285^{2} = 5221225, a palindromic square which consists of just 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 2286

(2286^{2} + 4) = (1^{2} + 4)(2^{2} + 4)(5^{2} + 4)(7^{2} + 4)(9^{2} + 4).

by Yoshio Mimura, Kobe, Japan

## 2288

2288 = (1^{2} + 2^{2} + 3^{2} + ... + 32^{2}) / (1^{2} + 2^{2}).

by Yoshio Mimura, Kobe, Japan

## 2291

2291^{k} + 2465^{k} + 2755^{k} + 5945^{k} are squares for k = 1,2,3 (116^{2}, 7366^{2}, 507964^{2}).

by Yoshio Mimura, Kobe, Japan

## 2293

2293^{4} = 27644976106801, and 2^{2} + 7^{2} + 6^{2} + 44^{2} + 9^{2} + 7^{2} + 6^{2} + 1^{2} + 0^{2} + 6^{2} + 8^{2} + 0^{2} + 1^{2} = 2293.

by Yoshio Mimura, Kobe, Japan

## 2294

2294^{2} = 5262436, a zigzag square.

2294^{2} = (11^{2} + 3)(206^{2} + 3).

by Yoshio Mimura, Kobe, Japan

## 2295

Komachi equations:

2295^{2} = 9^{2} * 8^{2} * 765^{2} / 4^{2} / 3^{2} / 2^{2} */ 1^{2} = 9^{2} / 8^{2} * 765^{2} * 4^{2} / 3^{2} * 2^{2} */ 1^{2}.

by Yoshio Mimura, Kobe, Japan

## 2296

2296^{2} = 5271616, a zigzag square.

2296^{2}± 3 are primes.

2296^{2} = 21^{3} + 64^{3} + 171^{3} = 56^{3} + 100^{3} + 160^{3}.

2296^{2} + 2297^{2} + 2298^{2} + ... + 5729^{2} = 5730^{2} + 5731^{2} + 5732^{2} + ... + 7140^{2}.

by Yoshio Mimura, Kobe, Japan

## 2297

2297^{2} = 16^{4} + 32^{4} + 34^{4} + 41^{4}.

by Yoshio Mimura, Kobe, Japan

## 2298

2298^{2} = 5280804, a zigzag square.

2298^{2}± 5 are primes.

by Yoshio Mimura, Kobe, Japan