Sums of digits & numbers of Row 2-9
00 11 33 05 30 54 88 20 46 65
00 11 33 50 03 45 88 02 64 56
11 × 32 = 352 , 253 = 23 × 11
2 × 32 = 64 , 46 = 23 × 2
19855 = 55 × 19 × 19 = 3^2 * 13^3 + 2*41.
5499535195 = 55 × 7 × 53 × 269519
55 × 31 × 1613 × 11443 × 3600407
==========
Row 2-1
64 42 88 10 97 56 65 93 34 46
= 2 × 13 × 101 × 1039763 × 23596687217
1 × 20) 100
2 × 10) 595 = 5 × 119
4 × _5) 35047 = 101 × 347
10 × 2) 12108814543 = 7 × 1729830649
644 = 1 × 92 × 7 , 446 = 2 × 223,
2+ 8+8+ 1+0+ 9+7+ 5+6+ 6+5+ 9+3+ 3 = 72
64428810975665933446 = 2 × 13 × 101 × 1039763 × 23596687217
56 65 > Palindrom
=======
Row 2-2 > 9 + 11 = 119
12 84 75 64 82 33 78 67 83 16
= 12 × 49 × 59 × 433 × 1327 × 6510247 × 9 × 11
1 × 20) 99 = 9 × 11
2 × 10) 594 = 6 × 9 × 11 = 2 × 297
4 × _5) 33264 = 16 × 21 × 9 × 11 = 5^2 *11^3 - 11.
10 × 2) 4663434798 = 2 × 17 × 103 × 13451 × 9 × 11
12847564823378678316 = 12 × 49 × 59 × 433 × 1327 × 6510247 × 9 × 11
594 is the smallest number with 10 representations as a sum of 3 positive squares:
594 = 1^2 + 8^2 + 23^2 = 3^2 + 3^2 + 24^2 = 3^2 + 12^2 + 21^2 = 4^2 + 7^2 + 23^2 = 4^2 + 17^2 + 17^2 = 5^2 + 13^2 + 20^2 = 7^2 + 16^2 + 17^2 = 8^2 + 13^2 + 19^2 = 12^2 + 15^2 + 15^2 = 13^2 + 13^2 + 16^2
==========
Row 2-3 (23=PR#5)
52 71 20 19 09 14 56 48 56 69
90 = 2 × 3 × 3 × 5 (23=PR#5)
414 = 6 × 69
19521 = 9 × 9 × 241 (PR#49) (2401=7^4)
6727687578 = 6 × 9 × 37 × 3367211
9 × 69 × 431 × 701 × 1543 × 182077733 (1543=PR#239)
52+ 71+ 20+ 19+ 09 = 171 = 9 × 19 , 143 = 11 × 13
14 +56 +48 +56 +69 = 243 = 3 × 3 × 3 × 3 × 3
5271201909 = 3 × 3 × 617 × 949253
1456485669 = 3 × 3 × 241 × 671501
19521 = 36^2 + 135^2
=========
Row 2-4
23 46 03 48 61 04 54 32 66 48
2 × 4 × 7 × 11 × 11 × 16747 × 206739777059
385 = 5 × 7 × 11 // 79
20878 = 2 × 11 × 13 × 73
2800361509 = 11 × 254578319
16747 = PR#1932 , 1932 = 12 × 7 × 23
6104 = 2 × 4 × 7 × 109
20878 = 2^3 *51^2 + 70.
==========
Row 2-5
21 33 93 60 72 60 24 91 41 27
27 × 647 × 1013 × 2099 × 574501909
522 = 2 × 9 × 29 / 72 = 2 × 2 × 2 × 9
25371 = 9 × 2819 = 11 *2^8 *3^2 + 3^3.
8158850199 = 27 × 302179637
25371 has the representation 25371
21 33 93 60 72 > 1972
23336 = 2 × 2 × 2 × 2917
===========
Row 2-6
37 24 58 70 06 60 63 15 52 54
2 × 17 × 2893129 × 378644286739
439 = PR#81= 2^9 - 73. / 79
21823 = 139 × 157 = 10 3^7 - 47.
9787742260 = 20 × 1523 × 321331
70+ 06+ 60+ 63 = 199 // 28
70*06*60*63 =1587600 = 6 × 6 × 6 × 6 × 5 × 5 × 7 × 7
37+ 24+ 58+ 15+ 52+ 54 = 240 // 51
37 *24 *58*15* 52* 54 = 2169348480 = 26 × 6^5 × 29 × 370
=========
Row 2-7
09 17 15 36 43 67 89 25 98 81
3 × 8311 × 367847288203957
480 // 102 = 2×3×17 = 2^7 - 26.
25626 = 2×3 × 4271 = 2^10 *5^2 + 26.
7706413524 = 12 × 89 × 101 × 71443
9 * 8 = 8 * 9 = 72 , 89 + 98 = 187 = 11 × 17
480 = 2^9 - 32.
=============
Row 2-8
74 88 15 20 92 09 62 82 90 36
2 × 2 × 67 × 1567 × 178308015413431
568 = 8 × 71 , 91 = 7 × 13
33535 = 5 × 19 × 353 = 2 *7^5 - 79.
8450981128 = 8 × 487 × 2169143
568 = 10^2 + 12^2 + 18^2 = 2^9 + 56.
=======
Row 2-9
00 11 33 05 30 54 88 20 46 65
5 × 11 × 31 × 1613 × 11443 × 3600407
352 = 2^8 + 2^4×6. / 64 = 2^6
19855 = 5 × 11 × 19 × 19 = 3^2 * 13^3 + 82.
5 × 7 × 11 × 53 × 269519
(1972 <> 710507)
======
Row 2-10
21 38 41 46 95 19 41 51 16 09
538739 × 39692962551331
337 = PR#64 // 80
21563 = PR#2416 = 3^2 * 7^4 - 2*23.
4079926304 = 2^5 × 127497697
38 41 46 95 19 41 51 16 = 347 // 68
(2 × 2 × 31 × 2320837 × 13348457)
337 = 9^2 + 16^2 = 2^8 + 81.
337^2 = 175^2 + 288^2
=====
22022020 = 172 × 29 × 5 × 883
20200222 = 2 × 7 × 1442873
========
Column 2-1
64+12+52+23+21+37+09+74+00+21
120223 × 533385651778027 (PR#11317)
313 = PR#61 = 12^2 + 13^2
61 = PR#14
14767 = PR#1726 = 3 * 17^3 + 28.
(2 × 2 × 2267 × 10399) PR#332, 1270
313^2 = 25^2 + 312^2
==========
Column 2-2
42+84+71+46+33+24+17+88+11+38
2 × 18397 × 1164514495092077
454 = 2 × 227
85 = 17 * 5
17680 = 80 × 13 × 17 = 16^2 + 132^2 = 36^2 + 128^2 = 48^2 + 124^2 = 92^2 + 96^2 = 2^3 *47^2 + 8.
6702595771 = PR
==========
Column 2-3
88+75+20+03+93+58+15+15+33+41
(3 × 3 × 31 × 318107540988584779)
441= 3 × 3 × 7 × 7
81 = 3*3*3*3
25092 = 36 × 17 × 41
= 66^2 + 144^2 = 96^2 + 126^2 = 2^3 *5^5 + 4*23.
14690353734 = 2 × 3 × 3 × 7 × 116590109
==========
Column 2-4
10+64+19+48+60+70+36+20+05+46
2 × 3 × 3 × 89 × 20757427 × 320025899
378 = 42 × 3 × 3 = 3 * 2^7 - 3 * 2.
72 = 24*3.
13248 = 24 × 24 × 23 = 13 * 2^10 - 2^6.
8100395406 = 2 × 3 × 3 × 13933 × 32299.
============
Column 2-5
97+82+09+61+72+06+43+92+30+95
5 × 179 × 27299 × 750353 × 5335763
587 = PR#103 = 588 - 1 , 587 = 2^9 + 75.
92 = 2 × 2 × 23
25436 = 2 × 2 × 6359 = 11 2^8 3^2 + 92.
10426019267 = 17 × 4241 × 144611
==========
Column 2-6
56+33+14+04+60+60+67+09+54+19
613 × 91894624153453663
376 = 8 × 47 = 3 * 2^7 - 8.
79 = PR#18
25225 = 5 × 5 × 1009=PR#165
25225 = 24^2 + 157^2 = 67^2 + 144^2 = 75^2 + 140^2
25225^2 = 7536^2 + 24073^2 = 16247^2 + 19296^2
11700235879 = 58013 × 201683
613^2 = 35^2 + 612^2
613 = 17^2 + 18^2
=============
Column 2-7
65+78+56+54+24+63+89+62+88+41
28456289 × 2311814244169
620 // 107 = PR#24
32498 = 2×16249
32498 = 5^2 6^4 + 98.
32498 = 77^2 + 163^2
12968194265 = 5 × 13 × 199510681
12968194265 is the hypotenuse of 4 primitive Pythagorean triples:
12968194265^2 = 1745477703^2 + 12850189496^2
= 1922394464^2 + 12824915673^2
= 7665548103^2 + 10460087704^2
= 7895330727^2 + 10287750736^2
===========
Column 2-8
93+67+48+32+91+15+25+82+20+51
10883×8607445824820897
524 = 4 * 131 = PR#28 , 83 = PR#19
27947 = PR#3046 = 7 * 2^5 * 5^3 - 53.
10893305342 = 2×5446652671
10883 = 2^5 7^3 - 93.
==============
Column 2-9
34+83+56+66+41+52+98+90+46+16
8×3×23×71×888846356789623
582 = 2×3×97 = 2^9 + 70. // 96 = 3 * 32 = 4^2 + 4^2 + 8^2
27807 = 3×13×23×31 = 3^4 * 7^3 + 3*2^3.
8782471257 = 3×7×418212917
=========
Column 2-10 , 2×3×5×7
46+16+69+48+27+54+81+36+65+09
11×1184475109×3543337691
451 = 11 * 41 = 2^9 - 61. // 100
28963 = 11×2633 = 170^2 + 63.
10098061336 = 8×11×17×71×95071
========
313+454+441+378+507 = 2093 = 7 × 13 × 23
376+620+524+582+451 = 2553 = 3 × 37 × 23
00 11 33 05 30 54 88 20 46 65
00 11 33 50 03 45 88 02 64 56
11 × 32 = 352 , 253 = 23 × 11
2 × 32 = 64 , 46 = 23 × 2
19855 = 55 × 19 × 19 = 3^2 * 13^3 + 2*41.
5499535195 = 55 × 7 × 53 × 269519
55 × 31 × 1613 × 11443 × 3600407
==========
Row 2-1
64 42 88 10 97 56 65 93 34 46
= 2 × 13 × 101 × 1039763 × 23596687217
1 × 20) 100
2 × 10) 595 = 5 × 119
4 × _5) 35047 = 101 × 347
10 × 2) 12108814543 = 7 × 1729830649
644 = 1 × 92 × 7 , 446 = 2 × 223,
2+ 8+8+ 1+0+ 9+7+ 5+6+ 6+5+ 9+3+ 3 = 72
64428810975665933446 = 2 × 13 × 101 × 1039763 × 23596687217
56 65 > Palindrom
=======
Row 2-2 > 9 + 11 = 119
12 84 75 64 82 33 78 67 83 16
= 12 × 49 × 59 × 433 × 1327 × 6510247 × 9 × 11
1 × 20) 99 = 9 × 11
2 × 10) 594 = 6 × 9 × 11 = 2 × 297
4 × _5) 33264 = 16 × 21 × 9 × 11 = 5^2 *11^3 - 11.
10 × 2) 4663434798 = 2 × 17 × 103 × 13451 × 9 × 11
12847564823378678316 = 12 × 49 × 59 × 433 × 1327 × 6510247 × 9 × 11
594 is the smallest number with 10 representations as a sum of 3 positive squares:
594 = 1^2 + 8^2 + 23^2 = 3^2 + 3^2 + 24^2 = 3^2 + 12^2 + 21^2 = 4^2 + 7^2 + 23^2 = 4^2 + 17^2 + 17^2 = 5^2 + 13^2 + 20^2 = 7^2 + 16^2 + 17^2 = 8^2 + 13^2 + 19^2 = 12^2 + 15^2 + 15^2 = 13^2 + 13^2 + 16^2
==========
Row 2-3 (23=PR#5)
52 71 20 19 09 14 56 48 56 69
90 = 2 × 3 × 3 × 5 (23=PR#5)
414 = 6 × 69
19521 = 9 × 9 × 241 (PR#49) (2401=7^4)
6727687578 = 6 × 9 × 37 × 3367211
9 × 69 × 431 × 701 × 1543 × 182077733 (1543=PR#239)
52+ 71+ 20+ 19+ 09 = 171 = 9 × 19 , 143 = 11 × 13
14 +56 +48 +56 +69 = 243 = 3 × 3 × 3 × 3 × 3
5271201909 = 3 × 3 × 617 × 949253
1456485669 = 3 × 3 × 241 × 671501
19521 = 36^2 + 135^2
=========
Row 2-4
23 46 03 48 61 04 54 32 66 48
2 × 4 × 7 × 11 × 11 × 16747 × 206739777059
385 = 5 × 7 × 11 // 79
20878 = 2 × 11 × 13 × 73
2800361509 = 11 × 254578319
16747 = PR#1932 , 1932 = 12 × 7 × 23
6104 = 2 × 4 × 7 × 109
20878 = 2^3 *51^2 + 70.
==========
Row 2-5
21 33 93 60 72 60 24 91 41 27
27 × 647 × 1013 × 2099 × 574501909
522 = 2 × 9 × 29 / 72 = 2 × 2 × 2 × 9
25371 = 9 × 2819 = 11 *2^8 *3^2 + 3^3.
8158850199 = 27 × 302179637
25371 has the representation 25371
21 33 93 60 72 > 1972
23336 = 2 × 2 × 2 × 2917
===========
Row 2-6
37 24 58 70 06 60 63 15 52 54
2 × 17 × 2893129 × 378644286739
439 = PR#81= 2^9 - 73. / 79
21823 = 139 × 157 = 10 3^7 - 47.
9787742260 = 20 × 1523 × 321331
70+ 06+ 60+ 63 = 199 // 28
70*06*60*63 =1587600 = 6 × 6 × 6 × 6 × 5 × 5 × 7 × 7
37+ 24+ 58+ 15+ 52+ 54 = 240 // 51
37 *24 *58*15* 52* 54 = 2169348480 = 26 × 6^5 × 29 × 370
=========
Row 2-7
09 17 15 36 43 67 89 25 98 81
3 × 8311 × 367847288203957
480 // 102 = 2×3×17 = 2^7 - 26.
25626 = 2×3 × 4271 = 2^10 *5^2 + 26.
7706413524 = 12 × 89 × 101 × 71443
9 * 8 = 8 * 9 = 72 , 89 + 98 = 187 = 11 × 17
480 = 2^9 - 32.
=============
Row 2-8
74 88 15 20 92 09 62 82 90 36
2 × 2 × 67 × 1567 × 178308015413431
568 = 8 × 71 , 91 = 7 × 13
33535 = 5 × 19 × 353 = 2 *7^5 - 79.
8450981128 = 8 × 487 × 2169143
568 = 10^2 + 12^2 + 18^2 = 2^9 + 56.
=======
Row 2-9
00 11 33 05 30 54 88 20 46 65
5 × 11 × 31 × 1613 × 11443 × 3600407
352 = 2^8 + 2^4×6. / 64 = 2^6
19855 = 5 × 11 × 19 × 19 = 3^2 * 13^3 + 82.
5 × 7 × 11 × 53 × 269519
(1972 <> 710507)
======
Row 2-10
21 38 41 46 95 19 41 51 16 09
538739 × 39692962551331
337 = PR#64 // 80
21563 = PR#2416 = 3^2 * 7^4 - 2*23.
4079926304 = 2^5 × 127497697
38 41 46 95 19 41 51 16 = 347 // 68
(2 × 2 × 31 × 2320837 × 13348457)
337 = 9^2 + 16^2 = 2^8 + 81.
337^2 = 175^2 + 288^2
=====
22022020 = 172 × 29 × 5 × 883
20200222 = 2 × 7 × 1442873
========
Column 2-1
64+12+52+23+21+37+09+74+00+21
120223 × 533385651778027 (PR#11317)
313 = PR#61 = 12^2 + 13^2
61 = PR#14
14767 = PR#1726 = 3 * 17^3 + 28.
(2 × 2 × 2267 × 10399) PR#332, 1270
313^2 = 25^2 + 312^2
==========
Column 2-2
42+84+71+46+33+24+17+88+11+38
2 × 18397 × 1164514495092077
454 = 2 × 227
85 = 17 * 5
17680 = 80 × 13 × 17 = 16^2 + 132^2 = 36^2 + 128^2 = 48^2 + 124^2 = 92^2 + 96^2 = 2^3 *47^2 + 8.
6702595771 = PR
==========
Column 2-3
88+75+20+03+93+58+15+15+33+41
(3 × 3 × 31 × 318107540988584779)
441= 3 × 3 × 7 × 7
81 = 3*3*3*3
25092 = 36 × 17 × 41
= 66^2 + 144^2 = 96^2 + 126^2 = 2^3 *5^5 + 4*23.
14690353734 = 2 × 3 × 3 × 7 × 116590109
==========
Column 2-4
10+64+19+48+60+70+36+20+05+46
2 × 3 × 3 × 89 × 20757427 × 320025899
378 = 42 × 3 × 3 = 3 * 2^7 - 3 * 2.
72 = 24*3.
13248 = 24 × 24 × 23 = 13 * 2^10 - 2^6.
8100395406 = 2 × 3 × 3 × 13933 × 32299.
============
Column 2-5
97+82+09+61+72+06+43+92+30+95
5 × 179 × 27299 × 750353 × 5335763
587 = PR#103 = 588 - 1 , 587 = 2^9 + 75.
92 = 2 × 2 × 23
25436 = 2 × 2 × 6359 = 11 2^8 3^2 + 92.
10426019267 = 17 × 4241 × 144611
==========
Column 2-6
56+33+14+04+60+60+67+09+54+19
613 × 91894624153453663
376 = 8 × 47 = 3 * 2^7 - 8.
79 = PR#18
25225 = 5 × 5 × 1009=PR#165
25225 = 24^2 + 157^2 = 67^2 + 144^2 = 75^2 + 140^2
25225^2 = 7536^2 + 24073^2 = 16247^2 + 19296^2
11700235879 = 58013 × 201683
613^2 = 35^2 + 612^2
613 = 17^2 + 18^2
=============
Column 2-7
65+78+56+54+24+63+89+62+88+41
28456289 × 2311814244169
620 // 107 = PR#24
32498 = 2×16249
32498 = 5^2 6^4 + 98.
32498 = 77^2 + 163^2
12968194265 = 5 × 13 × 199510681
12968194265 is the hypotenuse of 4 primitive Pythagorean triples:
12968194265^2 = 1745477703^2 + 12850189496^2
= 1922394464^2 + 12824915673^2
= 7665548103^2 + 10460087704^2
= 7895330727^2 + 10287750736^2
===========
Column 2-8
93+67+48+32+91+15+25+82+20+51
10883×8607445824820897
524 = 4 * 131 = PR#28 , 83 = PR#19
27947 = PR#3046 = 7 * 2^5 * 5^3 - 53.
10893305342 = 2×5446652671
10883 = 2^5 7^3 - 93.
==============
Column 2-9
34+83+56+66+41+52+98+90+46+16
8×3×23×71×888846356789623
582 = 2×3×97 = 2^9 + 70. // 96 = 3 * 32 = 4^2 + 4^2 + 8^2
27807 = 3×13×23×31 = 3^4 * 7^3 + 3*2^3.
8782471257 = 3×7×418212917
=========
Column 2-10 , 2×3×5×7
46+16+69+48+27+54+81+36+65+09
11×1184475109×3543337691
451 = 11 * 41 = 2^9 - 61. // 100
28963 = 11×2633 = 170^2 + 63.
10098061336 = 8×11×17×71×95071
========
313+454+441+378+507 = 2093 = 7 × 13 × 23
376+620+524+582+451 = 2553 = 3 × 37 × 23
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