## 9700

9700^{2} = 40^{4} + 60^{4} + 60^{4} + 90^{4}.

9700^{2} = 31 x 32 + 32 x 33 + 33 x 34 + ... + 655 x 656.

510^{k} + 690^{k} + 1330^{k} + 9570^{k} are squares for k = 1,2,3 (110^{2}, 9700^{2}, 937700^{2}).

by Yoshio Mimura, Kobe, Japan

## 9701

9701^{2} = 94109401, and 9^{2} + 4^{2} + 1^{2} + 0^{2} + 9^{2} + 4^{2} + 0^{2} + 1^{2} = 14^{2}.

by Yoshio Mimura, Kobe, Japan

## 9702

9702^{2} = 11^{4} + 59^{4} + 79^{4} + 81^{4}.

by Yoshio Mimura, Kobe, Japan

## 9703

9703^{2} = 43^{3} + 291^{3} + 411^{3}.

The square root of 9703 is 98.5038...., and 98 = 5^{2} + 0^{2} + 3^{2} + 8^{2}.

by Yoshio Mimura, Kobe, Japan

## 9704

9704^{2} = 94167616, and 9^{2} + 4^{2} + 1^{2} + 6^{2} + 7^{2} + 6^{2} + 1^{2} + 6^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 9705

9705^{2} = 94187025, a square with different digits.

by Yoshio Mimura, Kobe, Japan

## 9706

9706^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 9707

9707^{5} = 86183702586115165307 :

8^{2} + 6^{2} + 18^{2} + 37^{2} + 0^{2} + 2^{2} + 58^{2} + 6^{2} + 1^{2} + 15^{2} + 1^{2} + 65^{2} + 3^{2} + 0^{2} + 7^{2} = 9707.

by Yoshio Mimura, Kobe, Japan

## 9708

9708^{2} = S_{2}(215) + S_{2}(648), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + ... + n^{2}.

by Yoshio Mimura, Kobe, Japan

## 9709

9709^{2} = 94264681, and 94 * 26 * 4 - 68 + 1 = 9709.

9709^{5} = 86272524285864367549 :

8^{2} + 62^{2} + 7^{2} + 25^{2} + 24^{2} + 2^{2} + 8^{2} + 5^{2} + 8^{2} + 6^{2} + 4^{2} + 36^{2} + 7^{2} + 54^{2} + 9^{2} = 9709.

by Yoshio Mimura, Kobe, Japan

## 9710

9710^{2} = 92^{3} + 121^{3} + 451^{3}.

by Yoshio Mimura, Kobe, Japan

## 9712

9712^{2}± 3 are primes.

9712^{2} = 24^{4} + 44^{4} + 48^{4} + 96^{4}.

by Yoshio Mimura, Kobe, Japan

## 9715

9715^{2} = 94381225 appears in the decimal expressions of e:

e = 2.71828•••94381225••• (from the 30781st digit)

(94381225 is the sixth 8-digit square in the expression of e.)

by Yoshio Mimura, Kobe, Japan

## 9716

9716^{2} = 38^{3} + 126^{3} + 452^{3}.

by Yoshio Mimura, Kobe, Japan

## 9717

9717^{5} = 86628543710163888357 :

8^{2} + 66^{2} + 28^{2} + 5^{2} + 4^{2} + 37^{2} + 1^{2} + 0^{2} + 16^{2} + 38^{2} + 8^{2} + 8^{2} + 35^{2} + 7^{2} = 9717.

by Yoshio Mimura, Kobe, Japan

## 9720

117^{2} + 9720 = 153^{2}, 117^{2} - 9720 = 63^{2}.

by Yoshio Mimura, Kobe, Japan

## 9721

9721^{2} = 94497841, and 9^{2} + 4^{2} + 4^{2} + 9^{2} + 7^{2} + 8^{2} + 4^{2} + 1^{2} = 18^{2}.

9721^{5} = 86806993631541312601 :

8^{2} + 6^{2} + 80^{2} + 6^{2} + 9^{2} + 9^{2} + 3^{2} + 6^{2} + 3^{2} + 1^{2} + 54^{2} + 1^{2} + 3^{2} + 1^{2} + 2^{2} + 6^{2} + 0^{2} + 1^{2} = 9721.

by Yoshio Mimura, Kobe, Japan

## 9722

9722^{2} = 94517284, a zigzag square.

9722^{2} = 94517284, and 9^{2} + 4^{2} + 5^{2} + 1^{2} + 7^{2} + 2^{2} + 8^{2} + 4^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 9723

9723^{2}± 2 are primes.

1 / 9723 = 0.00010284891494, 10^{2} + 2^{2} + 84^{2} + 8^{2} + 9^{2} + 1^{2} + 49^{2} + 4^{2} = 9723.

by Yoshio Mimura, Kobe, Japan

## 9724

1326^{k} + 2730^{k} + 5226^{k} + 7618^{k} are squares for k = 1,2,3 (130^{2}, 9724^{2}, 779428^{2}).

by Yoshio Mimura, Kobe, Japan

## 9727

9727^{2} = 94614529, and 9 * 4 * 6 * 1 * 45 - 2 + 9 = 9727.

9727^{5} = 87075219795193937407 :

8^{2} + 7^{2} + 0^{2} + 75^{2} + 2^{2} + 19^{2} + 7^{2} + 9^{2} + 5^{2} + 19^{2} + 3^{2} + 9^{2} + 37^{2} + 40^{2} + 7^{2} = 9727.

by Yoshio Mimura, Kobe, Japan

## 9728

9728^{2} = 38^{3} + 346^{3} + 376^{3}.

by Yoshio Mimura, Kobe, Japan

## 9729

9729^{2}± 2 are primes.

9729^{5} = 87164775706363739649 :

8^{2} + 7^{2} + 16^{2} + 47^{2} + 7^{2} + 5^{2} + 7^{2} + 0^{2} + 63^{2} + 6^{2} + 37^{2} + 39^{2} + 6^{2} + 4^{2} + 9^{2} = 9729.

by Yoshio Mimura, Kobe, Japan

## 9733

1 / 9733 = 0.0001027432446316, 10^{2} + 2^{2} + 74^{2} + 32^{2} + 4^{2} + 46^{2} + 31^{2} + 6^{2} = 9733.

by Yoshio Mimura, Kobe, Japan

## 9736

9736 = (1^{2} + 2^{2} + 3^{2} + ... + 608^{2}) / (1^{2} + 2^{2} + 3^{2} + ... + 28^{2}).

by Yoshio Mimura, Kobe, Japan

## 9738

9738^{2} = 94828644, and 9482 - 8 + 6 * 44 = 9738.

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

9738 - 10853 - 2874 - 6260 - ... - 8136 - 7857 - 9333 - 9738

(Note f(9738) = 97^{2} + 38^{2} = 10853, f(10853) = 1^{2} + 08^{2} + 53^{2} = 2874, etc. See 41)

by Yoshio Mimura, Kobe, Japan

## 9739

9739^{2} = 94848121, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 9742

9742^{2} = 293^{3} + 318^{3} + 335^{3}.

9742^{2} = 94906564, zigzag aquare.

by Yoshio Mimura, Kobe, Japan

## 9744

9744^{2} = 16^{4} + 60^{4} + 80^{4} + 80^{4}.

9744^{2} = 94945536, and 9^{2} + 4^{2} + 9^{2} + 4^{2} + 5^{2} + 5^{2} + 3^{2} + 6^{2} = 17^{2}.

9744^{5} = 87838796432853172224 :

8^{2} + 7^{2} + 8^{2} + 3^{2} + 87^{2} + 9^{2} + 6^{2} + 4^{2} + 3^{2} + 28^{2} + 5^{2} + 31^{2} + 7^{2} + 2^{2} + 2^{2} + 2^{2} + 4^{2}

= 87^{2} + 8^{2} + 3^{2} + 8^{2} + 7^{2} + 9^{2} + 6^{2} + 4^{2} + 3^{2} + 28^{2} + 5^{2} + 31^{2} + 7^{2} + 2^{2} + 2^{2} + 2^{2} + 4^{2} = 9744.

by Yoshio Mimura, Kobe, Japan

## 9745

1^{2} + 4^{2} + 7^{2} + ... + (3x + 1)^{2} + ... + 9745^{2} = 320739^{2}.

9745^{2} = 94965025, and 9 * 4 * 9 * 6 * 5 + 0 + 25 = 9745.

by Yoshio Mimura, Kobe, Japan

## 9746

9746^{4} = 9022058279754256, and

9^{2} + 0^{2} + 2^{2} + 2^{2} + 0^{2} + 5^{2} + 8^{2} + 2^{2} + 7^{2} + 97^{2} + 5^{2} + 4^{2} + 2^{2} + 5^{2} + 6^{2} = 9746.

by Yoshio Mimura, Kobe, Japan

## 9747

9747^{2} = 57^{3} + 456^{3}.

A cubic polynomial :

(X + 2820^{2})(X + 4240^{2})(X + 9747^{2}) = X^{3} + 10997^{2}X^{2} + 51053100^{2}X + 116542929600^{2}.

by Yoshio Mimura, Kobe, Japan

## 9748

9748^{2} = 58^{3} + 170^{3} + 448^{3}.

by Yoshio Mimura, Kobe, Japan

## 9750

9750^{2} = (1^{2} + 9)(2^{2} + 9)(4^{2} + 9)(171^{2} + 9) = (1^{2} + 9)(4^{2} + 9)(11^{2} + 9)(54^{2} + 9)

= (4^{2} + 9)(11^{2} + 9)(171^{2} + 9).

9750^{2} = 200^{3} + 325^{3} + 375^{3}.

9750^{2} = 95062500, 9 and 5062500 = 2250^{2}.

by Yoshio Mimura, Kobe, Japan

## 9754

The square root of 9754 is 98.7623...., and 98 = 7^{2} + 6^{2} + 2^{2} + 3^{2}.

9754^{5} = 88290455272414981024 :

8^{2} + 8^{2} + 29^{2} + 0^{2} + 4^{2} + 5^{2} + 5^{2} + 27^{2} + 24^{2} + 14^{2} + 9^{2} + 81^{2} + 0^{2} + 24^{2} = 9754.

by Yoshio Mimura, Kobe, Japan

## 9756

9756^{2} = 95179536, a square with odd digits except the last digit 6.

The square root of 9756 is 98.77...., and 98 = 7^{2} + 7^{2}.

9756^{2} = (13 + 14 + 15 + ... + 36)^{2} + (37 + 38 + 39 + ... + 60)^{2} + (61 + 62 + 63 + ... + 84)^{2} + ... + (205 + 206 + 207 + ... + 228)^{2}.

by Yoshio Mimura, Kobe, Japan

## 9757

The square root of 9757 is 98.77...., and 98 = 7^{2} + 7^{2}.

1 / 9757 = 0.00010249051962693, 10^{2} + 24^{2} + 9^{2} + 05^{2} + 19^{2} + 62^{2} + 69^{2} + 3^{2} = 9757.

1 / N = 0.0001024...., and 1024 = 32^{2}, for N = 9757, 9758,..., 9765.

9757^{3} = 928857121093, and 9^{2} + 28^{2} + 8^{2} + 5^{2} + 7^{2} + 1^{2} + 2^{2} + 10^{2} + 93^{2} = 9757.

by Yoshio Mimura, Kobe, Japan

## 9758

9758^{2} = 95218564, and 9 * 5 * 218 - 56 + 4 = 9758.

by Yoshio Mimura, Kobe, Japan

## 9759

9759^{2}± 2 are primes.

by Yoshio Mimura, Kobe, Japan

## 9760

9760^{5} = 88562341091737600000 :

8^{2} + 85^{2} + 6^{2} + 23^{2} + 41^{2} + 0^{2} + 9^{2} + 1^{2} + 7^{2} + 3^{2} + 7^{2} + 6^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} + 0^{2} = 9760.

by Yoshio Mimura, Kobe, Japan

## 9761

9761^{2} = 158^{3} + 266^{3} + 417^{3}.

by Yoshio Mimura, Kobe, Japan

## 9763

1 / 9763 = 0.00010242753252074, 10^{2} + 2^{2} + 4^{2} + 27^{2} + 53^{2} + 25^{2} + 2^{2} + 074^{2} = 9763.

by Yoshio Mimura, Kobe, Japan

## 9764

9764^{5} = 88743970114138317824 :

8^{2} + 8^{2} + 7^{2} + 4^{2} + 39^{2} + 7^{2} + 0^{2} + 1^{2} + 1^{2} + 4^{2} + 13^{2} + 8^{2} + 31^{2} + 7^{2} + 82^{2} + 4^{2} = 9764.

by Yoshio Mimura, Kobe, Japan

## 9765

9765^{2} = 242^{3} + 433^{3} = 49^{3} + 338^{3} + 384^{3}.

9765^{2} = 95355225, and 9535 + 5 + 225 = 9765.

9765^{5} = 88789423898328103125 :

8^{2} + 8^{2} + 7^{2} + 8^{2} + 9^{2} + 4^{2} + 2^{2} + 3^{2} + 89^{2} + 8^{2} + 3^{2} + 28^{2} + 1^{2} + 0^{2} + 3^{2} + 1^{2} + 25^{2}

= 8^{2} + 8^{2} + 7^{2} + 89^{2} + 4^{2} + 2^{2} + 3^{2} + 8^{2} + 9^{2} + 8^{2} + 3^{2} + 28^{2} + 1^{2} + 0^{2} + 3^{2} + 1^{2} + 25^{2} = 9765.

by Yoshio Mimura, Kobe, Japan

## 9766

9766^{2} = 95374756, and 9537 + 47 * 5 - 6 = 9766.

by Yoshio Mimura, Kobe, Japan

## 9767

9767^{2} = 95394289, and 9^{2} + 5^{2} + 3^{2} + 9^{2} + 4^{2} + 2^{2} + 8^{2} + 9^{2} = 19^{2}.

by Yoshio Mimura, Kobe, Japan

## 9768

9768^{2} = 36^{3} + 176^{3} + 448^{3} = 49^{3} + 359^{3} + 366^{3}.

The integral triangle of sides 113, 1874080, 1874129 has square area 9768^{2}.

by Yoshio Mimura, Kobe, Japan

## 9770

The square root of 9770 is 98.8433...., and 98 = 8^{2} + 4^{2} + 3^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 9772

The square root of 9772 is 98.853, and 98 = 8^{2} + 5^{2} + 3^{2}.

by Yoshio Mimura, Kobe, Japan

## 9773

9773^{2} = 151^{3} + 205^{3} + 437^{3}.

by Yoshio Mimura, Kobe, Japan

## 9774

9774^{5} = 89199346672309542624 :

8^{2} + 9^{2} + 19^{2} + 9^{2} + 34^{2} + 66^{2} + 7^{2} + 2^{2} + 3^{2} + 0^{2} + 9^{2} + 54^{2} + 2^{2} + 6^{2} + 24^{2} = 9774.

by Yoshio Mimura, Kobe, Japan

## 9777

9777^{2} = 313^{3} + 318^{3} + 320^{3}.

9777^{2} = 95589729, and 9 * 5 - 5 + 8 + 9729 = 9777.

by Yoshio Mimura, Kobe, Japan

## 9778

9778^{2} = 95609284, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 9779

9779^{2} = 164^{3} + 176^{3} + 441^{3}.

by Yoshio Mimura, Kobe, Japan

## 9781

9781^{4} = 9152358761897521, and

9^{2} + 1^{2} + 5^{2} + 2^{2} + 3^{2} + 58^{2} + 7^{2} + 6^{2} + 1^{2} + 8^{2} + 9^{2} + 75^{2} + 21^{2} = 9781.

by Yoshio Mimura, Kobe, Japan

## 9784

The square root of 9784 is 98.914...., and 98 = 9^{2} + 1^{2} + 4^{2}.

by Yoshio Mimura, Kobe, Japan

## 9785

969^{k} + 5016^{k} + 5092^{k} + 6612^{k} are squares for k = 1,2,3 (133^{2}, 9785^{2}, 740411^{2}).

by Yoshio Mimura, Kobe, Japan

## 9786

9786^{2} = 95765796, and 95 * 7 * 6 + 5796 = 9786.

by Yoshio Mimura, Kobe, Japan

## 9789

9789^{2} = 95824521, and 9582 + 4 * 52 - 1 = 9789.

by Yoshio Mimura, Kobe, Japan

## 9791

9791^{2} = 95863681, and 9586 + 3 * 68 + 1 = 9791.

by Yoshio Mimura, Kobe, Japan

## 9792

9792^{2}± 5 are primes.

9792^{2} = (17^{2} - 1)(577^{2} - 1) = (2^{2} - 1)(5^{2} - 1)(33^{2} - 1)(35^{2} - 1).

9792^{2} = 16^{3} + 288^{3} + 416^{3} = 16^{4} + 32^{4} + 56^{4} + 96^{4} = 24^{4} + 48^{4} + 48^{4} + 96^{4}.

Komachi equation: 9792^{2} = 1^{2} / 2^{2} * 34^{2} * 56^{2} / 7^{2} * 8^{2} * 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 9794

9794^{2} = 95922436, and 9^{2} + 5^{2} + 9^{2} + 2^{2} + 2^{2} + 4^{2} + 3^{2} + 6^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 9799

The sum of the squares of divisors of 9799 is a square, 9802^{2}.

by Yoshio Mimura, Kobe, Japan