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.
==========
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
=========