Square 1

31+ 41+ 59+ 26+ 53+ 58+ 97+ 93+ 23+ 84




97 = PR#21 = 68 + 29 = 2 × 2 × 17 + 29

565 = 5 × 113 = 365 + 200 = 7 * 75 + 10 = 7 × 105 - 170 × 1

113 = PR#26

565 = 7 + 558 = 7 + 2 × 3 × 3 × 31 = 7 + 2 × 279 × 1

26602 = 2×47×283, = 13 * 2^11 - 2 * 11.

9039525037 = 103×6703×13093

2^5×563×15647×93967×1186001




31+41+59+53+97+23 = 304 = 2^4×19 = 2^8 + 48.

304 has the unique representation 304 = 4^2 + 12^2 + 12^2 as a sum of 3 squares.

565 - 304 = 261 = 9×29 = 6^2 + 15^2 = 256 + 5.

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62 64 33 83 27 95 02 88 41 97




592 = 2^4 × 37 = 4^2 + 24^2 = 2^9 + 2^4 × 5.

97 = PR#21 = 68 + 29 = 2 × 2 × 17 + 29.

16927 = 27 × 25 × 25 + 52.

16928 = 2 × 92 × 92 = 32 × 23 × 23.

15767222524 = 2^2 × 1303 × 3025177.

187651 × 3325811 × 100375277.




62+64+33+83+27+95 = 364 = 2 × 7 × 26 = 2^2 + 6^2 + 18^2.

62+64+33+83+27 = 269 = PR#53 = 10^2 + 13^2.

269^2 = 69^2 + 260^2 = 270 - 1.

58 = 2×29




95+02+88+41+97 = 323 = 17 × 19

41 = PR#9




269 + 565 = 834 = 6 × 139. (2*2*2*17)

58+97 = 5×31 = 155 = 2^7 + 27.




==========

16 93 99 37 51 05 82 09 74 94

98×863×227497×880438973

560 = 4^2 + 12^2 + 20^2 // 101 = PR#22

32438 = 5^2 × 6^4 + 38 = 98×331

2276091245 = 5×29×15697181

2276091245 has 4 representations as a sum of 2 squares:

2276091245 = 14842^2 + 45341^2 = 15331^2 + 45178^2 = 20522^2 + 43069^2 = 22142^2 + 42259^2

2276091245 is the hypotenuse of 4 primitive Pythagorean triples:

2276091245^2 = 1295554917^2 + 1871397556^2 = 1345902244^2 + 1835521317^2 = 1385247836^2 + 1806012123^2 = 1433786277^2 + 1767724036^2

169399375 1 0 582097494

169399375 = 52 , 582097494 = 48

Първата десятка и второто 38 и точен квадрат 16.

============

45 92 30 78 16 40 62 86 20 89

9×41×5323×917281×25488587

558 = 18×31 = 600 - 42 // 90 = 3^2 + 9^2 = 4^2 + 5^2 + 7^2

90 is the smallest number with 6 representations as a sum of 4 positive squares:

90 = 1^2 + 2^2 + 2^2 + 9^2 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2 = 2^2 + 5^2 + 5^2 + 6^2 = 3^2 + 3^2 + 6^2 + 6^2 = 3^2 + 4^2 + 4^2 + 7^2

4+5+ 9+2+ 3+0+ 7+8+ (1+6+ 4+0)+ 6+2+ 8+6+ 2+0+ 8+9

38 + 11 + 41 = 45 + 45 = 90

17685 = 9×5×131

8655169905 = 9×5×23×41×59×3457

Row 4-5 (71050792...) Този ред и неговите суми показва ред 4=5.




=======

98 62 80 34 82 53 42 11 70 67.

7×194507×2113987×34266109.

599 = PR#105, // 86 = 2×43=PR#10.

37427 = 13×2879 = 12 ×125 - 73.

15204920549 = 7×7×13×23869577.

86 - 25 = 61=PR#14 , 599 - 160 = 439 = PR#81.

986 * 2 = 1972.

37427 - 9862 = 27565 = 5× 37 × 149 = 2^10× 3^3 - 83 =

27565 = 3^2 + 166^2 = 51^2 + 158^2 = 54^2 + 157^2=102^2 + 131^2.

27565 is the hypotenuse of 4 primitive Pythagorean triples:

27565^2 = 996^2 + 27547^2 = 6757^2 + 26724^2 =

16116^2 + 22363^2 = 16956^2 + 21733^2.

=========

98 21 48 08 65 13 28 23 06 64

44×149×7490452154616247

374 = 2×11×17 = 3 *2^7 - 10. // 86

24629 = 11×2239

11149711529 = 11×1013610139

374 is the smallest number with 8 representations as a sum of 3 positive squares:

374 = 1^2 + 7^2 + 18^2 = 2^2 + 3^2 + 19^2 = 2^2 + 9^2 + 17^2 = 3^2 + 13^2 + 14^2 = 5^2 + 5^2 + 18^2 = 6^2 + 7^2 + 17^2 = 6^2 + 13^2 + 13^2 = 7^2 + 10^2 + 15^2

11 във всички сборове показва ред 2-1 или 11 броено последователно, който започва със 6, а предходните две цифри са 19, 196 е първото корабно число. Редът също важността на всяка цифра и нейното място в таблицата.

==========

70 93 84 46 09 55 05 82 23 17

11×17×12269×30919388631539.

12599666926 = 2×7×11×17×4812707.

19393 = 11×41×43, 484 = 22^2 // 88 = 4^2 + 6^2 + 6^2.

93844 = 2^2×29×809(8+9=17) // 58223 = 11×67×79 (PR#15, 18).

32285 = 7 * 2^9 * 9 + 29.

70 93 84 46 09 = 302 / 50 ,, 55 05 82 23 17 = 182 / 38.

==========

25 35 94 08 12 84 81 11 74 50

2×3×5^2×557×303523735828619

11017058262 = 2×3×7×262310911

28788 = 2^2×3×2399 // 474 = 2×3×79 // 78 = 2×3×13




===========

28 41 02 70 19 38 52 11 05 55

3×5×29669×3838519×16630967.

321 = 3*107 // 69 = 3*23.

10815 = 3×5×7×103 = 2^6 * 13^2 - 1.

6693137574 = 2×3×1115522929

Третото павлово число е 69. Редът има и връзка със двете уникални числа.




===============

96 44 62 29 48 95 49 30 38 19

3×7×13×353282891903111003 (22)

510 = 2×3×5×17, 105 = 3×5×7

29517 = 3×9839 = PR#2110

19193926767 = 3×113×56619253 (PR#26)

105 , 510 > Ред 10 от общо 5 квадрата. Връзка с 3-ти кв. 51 05.

========

Columns




31+62+16+45+98+98+70+25+28+96

2^5×17×1709×2129×15975992519

13032417494 = 674×19335931

24626 = 2×7×1759




569 has a representation as a sum of 2 squares:

569 = PR#100 = 13^2 + 20^2 ,

569 is the hypotenuse of a primitive Pythagorean triple:

569^2 = 231^2 + 520^2

101 = PR#22

965 = 2^2 + 31^2 = 17^2 + 26^2 = 5×193=PR#40

965^2 = 124^2 + 957^2 = 387^2 + 884^2

31+62+16+45+98 = 252 = 36×7 // 45

98+70+25+28+96 = 317 = PR#62 = 11^2 + 14^2 // 56

317^2 = 75^2 + 308^2




1+2+6+5+8+8+0+5+8+6 = 49 ,, 101 - 49 = 52

===========

41+64+93+92+62+21+93+35+41+44

2^5×19×47×59×3823×6461761517.

586 = 2×293 = 15^2 + 19^2 = 2^9 + 74.

33256 = 2^3×4157 = 66^2 + 170^2 = 5^2 × 11^3 - 19.

6358293406 = 2 × 3179146703.

1+4+3+2+2+1+3+5+1+4 = 26 ,, 82 - 26 = 56.

82 is the smallest number with 5 representations as a sum of 4 positive squares:

82 = 1^2 + 1^2 + 4^2 + 8^2 = 1^2 + 3^2 + 6^2 + 6^2 = 1^2 + 4^2 + 4^2 + 7^2 = 2^2 + 2^2 + 5^2 + 7^2 = 4^2 + 4^2 + 5^2 + 5^2

==========

59+33+99+30+80+48+84+94+02+62

591 = 3×197 = 2^9 + 79. // 96 = 32×3 = 4^2 + 4^2 + 8^2

5+3+9+3+8+4+8+9+0+6 = 55 ,, 96 - 55 = 41

32667 = 3×10889

10818933342 = 2×3×7^2×2411×15263

2×3×13×37×123719×166193604743

=======

26+83+37+78+34+08+46+08+70+29

419 = PR#77, 95 = 5*19

2+8+3+7+3+0+4+0+7+2 = 36 , 95 - 36 = 59

21506 = 2×10753 = 91^2 + 115^2

3529464863 = 173×251×81281

7×107×3582613931703721

=========

53+27+51+16+82+65+09+12+19+48

382 = 2×191 ,, 85 = 5 * 17 = 2^2 + 9^2 = 6^2 + 7^2

5+2+5+1+8+6+0+1+1+4 = 33 ,, 85 - 33 = 52

21568 = 2^6×337 = 72^2 + 128^2

11836633630 = 530×2281×9791

2^2×71×187588439529961697




85^2 = 13^2 + 84^2 = 36^2 + 77^2

=============

58+95+05+40+53+13+55+84+38+95

536 = 8×67, 95 = 5*19

5+9+0+4+5+1+5+8+3+9 = 49 + 46 = 95

21227 = PR#2383

7250897948 = 2^2×7×821×315421

5×7×47×73×490906778792987

=========

97+02+82+62+42+28+05+81+52+49

500 = 4^2 + 22^2 = 10^2 + 20^2 ! // 86

9+0+8+6+4+2+0+8+5+4 = 46 + 40 = 86

28022 = 2×14011 ,, 9988641491 = PR

13×75883513×9835749821

===========

93+88+09+86+11+23+82+11+11+30

444 = 2^9 - 68. // 75

9+8+0+8+1+2+8+1+1+3 = 41 + 34 = 75

20838 = 2^3 * 51^2 + 30 = 2×3×23×151

11770209741 = 3×20939×187373

2×3×5×107×1672753×17483999801




======

23+41+74+20+70+06+23+74+05+38

374 = 22×17 // 68 = 2^2 + 8^2

2+4+7+2+7+0+2+7+0+3 = 34 + 34 = 68

19679 = 11×1789 = 3^9 - 4.

2404116608 = 2^7×17×1104833

2×13^3×1073507×496448411




2404116608 = 6728^2 + 48568^2 = 28792^2 + 39688^2

374 is the smallest number with 8 representations as a sum of 3 positive squares:

374 = 1^2 + 7^2 + 18^2 = 2^2 + 3^2 + 19^2 = 2^2 + 9^2 + 17^2 = 3^2 + 13^2 + 14^2 = 5^2 + 5^2 + 18^2 = 6^2 + 7^2 + 17^2 = 6^2 + 13^2 + 13^2 = 7^2 + 10^2 + 15^2

========

84+97+94+89+67+64+17+50+55+19

636 = 12*53 = 6*105 = 4*159 = 2*318 // 114 = 19*6

8+9+9+8+6+6+1+5+5+1 = 58 + 56 = 114

32019 = 3×13×821 = 2^8 * 5^3 + 19.

14915454486 = 2×3×2485909081

3×7×37321×108428026192859




114 is the smallest number with 10 representations as a sum of 2 primes:

114 = 5 + 109 = 7 + 107 = 11 + 103 = 13 + 101 = 17 + 97 = 31 + 83 = 41 + 73 = 43 + 71 = 47 + 67 = 53 + 61

=========