Puzzle 1119 Consecutive primes as 11 & 13

Problems & Puzzles: Puzzles

Puzzle 1119 Consecutive primes as 11 & 13

JM Bergot sent the following nice puzzle:

For consecutive primes 11 and 13, 11^2 + 13^2 = 290, add 1 to get 291= 3*97 and 3+97=100=10^2.

Q. Others two consecutive primes like 11 and 13 (producing a power of 10)?

During the week from Jan 14-20 2023, contributions came from Martin Hopf, Emmanuel Vantieghem, Alain Rochelli, Michael Branicky, Gennady Gusev, Paul Cleary, J-M Rebert, Oscar Volpatti

***

Martin wrote:

My twin primes and the corresponding solutions base 10^n:

[p, p+2] --> solutions base 10^n
--------------------------------
[11, 13] --> 10^2
[189632411, 189632413] --> 10^4
[4597861451, 4597861453] --> 10^5
[9378919721, 9378919723] --> 10^7
[18690564071, 18690564073] --> 10^6
[21940293251, 21940293253] --> 10^5
[37978296617, 37978296619] --> 10^8
[67455481019, 67455481021] --> 10^4
[78288371177, 78288371179] --> 10^4
[79711456001, 79711456003] --> 10^4
[104342291699, 104342291701] --> 10^5
[155701545827, 155701545829] --> 10^5
[173041382621, 173041382623] --> 10^5

***

Emmanuel wrote:

The puzzle asks to find at least two odd primes p and q such that we can find an integer h such that :
p^2 + q^2 + 1 = h*(10^n - h). (*)

When n = 1 there is no solution, simply by exhausting..

When n = 2 there are 21 solutions, (also found by exhausting) which you can find in the next table :
p q h 100 - h
7 7 1 99
11 23 7 93
17 19 7 93 *
17 23 9 91
13 31 13 87
17 29 13 87
13 37 19 81
17 37 21 79
5 43 25 75
11 43 27 73
17 41 27 73
17 43 31 69
19 43 33 67
23 41 33 67
29 37 33 67
11 47 37 63
31 37 37 63 *
13 47 39 61
23 43 39 61
17 47 49 51
(marked with * : p, q consecutive;
marked with ** : p, q consecutive and h, 100 - h prime)

When n > 2 there are no solutions.
This is because the equation (*) mod 8 reads :
3 == -h^2 mod 8
which has no solutions.

***

Alain wrote:

It seems that there are no other consecutive primes which are solutions.

On the other hand, we can obtain solutions if we replace “power of 10” by “odd multiple of 100”.

In this case, the first suitable solutions are:

11 ; 13
137 ; 139
461 ; 463
541 ; 547
1933 ; 1949

***

Michael wrote:

I found further examples with lower consecutive prime values as follows:
56681, 4341083, 41512687, 44586709, 66103351, 91416967, 138018481, 168218471, 189632411, 190889693, 427396637

For consecutive primes 56681 and 56687:
56681^2 + 56687^2 = 6426151730,
add 1 to get 6426151731 = 3*3*29*179*263*523
and 3+3+29+179+263+523 = 1000 = 10^3

For consecutive primes 4341083 and 4341107:
4341083^2 + 4341107^2 = 37690211598338,
add 1 to get 37690211598339 = 3*19*67*467*3643*5801
and 3+19+67+467+3643+5801 = 10000 = 10^4

For consecutive primes 41512687 and 41512729:
41512687^2 + 41512729^2 = 3446609850987410,
add 1 to get 3446609850987411 = 3*79*719*1571*2521*5107
and 3+79+719+1571+2521+5107 = 10000 = 10^4

For consecutive primes 44586709 and 44586761:
44586709^2 + 44586761^2 = 3975953875921802,
add 1 to get 3975953875921803 = 3*29*61*1733*4621*93553
and 3+29+61+1733+4621+93553 = 100000 = 10^5

For consecutive primes 66103351 and 66103363:
66103351^2 + 66103363^2 = 8739307613338970,
add 1 to get 8739307613338971 = 3*101*1319*2207*2699*3671
and 3+101+1319+2207+2699+3671 = 10000 = 10^4

For consecutive primes 91416967 and 91416991:
91416967^2 + 91416991^2 = 16714128098973170,
add 1 to get 16714128098973171 = 3*419*499*2371*3251*3457
and 3+419+499+2371+3251+3457 = 10000 = 10^4

For consecutive primes 138018481 and 138018499:
138018481^2 + 138018499^2 = 38098207163760362,
add 1 to get 38098207163760363 = 3*3*139*18541*37423*43891
and 3+3+139+18541+37423+43891 = 100000 = 10^5

For consecutive primes 168218471 and 168218509:
168218471^2 + 168218509^2 = 56594920755760922,
add 1 to get 56594920755760923 = 3*59*571*613*10331*88423
and 3+59+571+613+10331+88423 = 100000 = 10^5

For consecutive primes 189632411 and 189632413:
189632411^2 + 189632413^2 = 71920903361875490,
add 1 to get 71920903361875491 = 3*7*7*7*11*31*73*419*773*8669
and 3+7+7+7+11+31+73+419+773+8669 = 10000 = 10^4

For consecutive primes 190889693 and 190889731:
190889693^2 + 190889731^2 = 72877764294886610,
add 1 to get 72877764294886611 = 3*59*263*1483*12289*85903
and 3+59+263+1483+12289+85903 = 100000 = 10^5

For consecutive primes 427396637 and 427396663:
427396637^2 + 427396663^2 = 365335792862445338,
add 1 to get 365335792862445339 = 3*11*317*25463*34847*39359
and 3+11+317+25463+34847+39359 = 100000 = 10^5

***

Gennady wrote:

Let S be the sum of squares of consecutive primes.

Сonsider the case when S+1 has 2 prime divisors.

Then S+1=p1*p2 and p1+p2=10^n.

We have p2=10^n-p1.

p1 cannot be of the form 3*k+1, otherwise p2 is a composite (10^n-3*k-1 is divisible by 3).

Well known that,

an integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor p^m, where prime p of the form 4*k+3 and m is odd.

So p1 cannot be of the form 3*k+2, otherwise S = (3*k+2)*(10^n-(3*k+2))-1, will be divisible by 3. And .S cannot be written as sum of 2 squares.

I.e. p1=3 is the only option.

We have

p1=3, p2=10^n-3. p2 is prime for n=2 (the puzzle condition), 3, 17, 140, 990, 1887, 3530...

n=3, p2=997, S=3*997-1=2*5*13*23, contains 23=4*k+3.

n=7, p2=9999997, S=3*9999997-1=2*5*2999999, 2999999=4*k+3.

n=140, p2=10^140-3, S has a divisor of 19.

n=990, S has a divisor of 7.

In all these cases, S cannot be the sum of the squares of any numbers.

The following n are too big for factorization.

So the example from the problem is the only solution for n<1887.

I did not consider the folowing variants of the problem:

the S+1 has more than 2 prime divisors - what should I sum up?

Sum up the distinct prime divisors, all divisors in general?

***

Paul wrote:

Here are some other consecutive primes that produce powers of 10.

Consecutive Prime 11 and 13 -> 11^2 + 13^2 + 1 = 291 = {3,97} = 100 = , 10^2

Consecutive Prime 56681 and 56687 -> 56681^2 + 56687^2 + 1 = 6426151731 = {3+3+29+179+263+523} = 1000 = , 10^3

Consecutive Prime 4341083 and 4341107 -> 4341083^2 + 4341107^2 + 1 = 37690211598339 = {3+19+67+467+3643+5801} = 10000 = , 10^4

Consecutive Prime 41512687 and 41512729 -> 41512687^2 + 41512729^2 + 1 = 3446609850987411 = {3+79+719+1571+2521+5107} = 10000 = , 10^4

Consecutive Prime 44586709 and 44586761 -> 44586709^2 + 44586761^2 + 1 = 3975953875921803 = {3+29+61+1733+4621+93553} = 100000 = , 10^5

Consecutive Prime 66103351 and 66103363 -> 66103351^2 + 66103363^2 + 1 = 8739307613338971 = {3+101+1319+2207+2699+3671} = 10000 = , 10^4

Consecutive Prime 91416967 and 91416991 -> 91416967^2 + 91416991^2 + 1 = 16714128098973171 = {3+419+499+2371+3251+3457} = 10000 = , 10^4

Consecutive Prime 138018481 and 138018499 -> 138018481^2 + 138018499^2 + 1 = 38098207163760363 = {3+3+139+18541+37423+43891} = 100000 = , 10^5

Consecutive Prime 168218471 and 168218509 -> 168218471^2 + 168218509^2 + 1 = 56594920755760923 = {3+59+571+613+10331+88423} = 100000 = , 10^5

Consecutive Prime 189632411 and 189632413 -> 189632411^2 + 189632413^2 + 1 = 71920903361875491 = {3+7+7+7+11+31+73+419+773+8669} = 10000 = , 10^4

Consecutive Prime 190889693 and 190889731 -> 190889693^2 + 190889731^2 + 1 = 72877764294886611 = {3+59+263+1483+12289+85903} = 100000 = , 10^5

Consecutive Prime 427396637 and 427396663 -> 427396637^2 + 427396663^2 + 1 = 365335792862445339 = {3+11+317+25463+34847+39359} = 100000 = , 10^5

Consecutive Prime 536753377 and 536753389 -> 536753377^2 + 536753389^2 + 1 = 576208388323889451 = {3+3+3+3+3+13+137+1459+10369+88007} = 100000 = , 10^5

Consecutive Prime 678782777 and 678782801 -> 678782777^2 + 678782801^2 + 1 = 921492149285237331 = {3+41+827+5419+23971+69739} = 100000 = , 10^5

Consecutive Prime 738896399 and 738896429 -> 738896399^2 + 738896429^2 + 1 = 1091935821244119243 = {3+7+7+7+7+229+229+503+691+8317} = 10000 = , 10^4

***

Jean-Marc wrote:

Q. Others two consecutive primes like 11 and 13 (producing a power of 10)?

Least d-digit solutions.
Let s be the sum of prime factors with multiplicity of p^2 + q^2 + 1 with p, q consecutive primes.

d, p, q, p^2 + q^2 + 1, s
2 11 13 291 100 = 10^2
5 56681 56687 6426151731 1000 = 10^3
7 4341083 4341107 37690211598339 10000 = 10^4
8 41512687 41512729 3446609850987411 10000 = 10^4
9 138018481 41512729 38098207163760363 100000 = 10^5
10 1031230009 1031230027 2126870700048560811 1000000 = 10^6
11 10011656233 10011656273 200466521856468000819 10000 = 10^4
12 101904986921 20769252719961221963571 10000000 = 10^7
13 1004175782717 2016738005194615894935051 100000 = 10^5
14 10018139795273 200726249915625745738919211 100000 = 10^5

Least 2-digit solutions :
For consecutive primes 11 and 13, 11^2 + 13^2 = 290, add 1 to get 291 = (3 * 97) and 3 + 97 = 100 = 10^2.

Least 5-digit :
For consecutive primes 56681 and 56687, 56681^2 + 56687^2 = 6426151730, add 1 to get 6426151731 = (3^2 * 29 * 179 * 263 * 523) and 3*2 + 29 + 179 + 263 + 523 = 1000 = 10^3.

Least 7-digit :
For consecutive primes 4341083 and 4341107, 4341083^2 + 4341107^2 = 37690211598338, add 1 to get 37690211598339 = (3 * 19 * 67 * 467 * 3643 * 5801) and 3 + 19 + 67 + 467 + 3643 + 5801 = 10000 = 10^4.

Least 8-digit :
For consecutive primes 41512687 and 41512729, 41512687^2 + 41512729^2 = 3446609850987410, add 1 to get 3446609850987411 = (3 * 79 * 719 * 1571 * 2521 * 5107) and 3 + 79 + 719 + 1571 + 2521 + 5107 = 10000 = 10^4.

Least 9-digit :
For consecutive primes 138018481 and 138018499, 138018481^2 + 138018499^2 = 38098207163760362, add 1 to get 38098207163760363 = (3^2 * 139 * 18541 * 37423 * 43891) and 3*2 + 139 + 18541 + 37423 + 43891 = 100000 = 10^5.

Least 10-digit
For consecutive primes 1031230009 and 1031230027, 1031230009^2 + 1031230027^2 = 2126870700048560810, add 1 to get 2126870700048560811 = (3 * 19 * 317 * 3539 * 34591 * 961531) and 3 + 19 + 317 + 3539 + 34591 + 961531 = 1000000 = 10^6.

Least 11-digit
For consecutive primes 10011656233 and 10011656273, 10011656233^2 + 10011656273^2 = 200466521856468000818, add 1 to get 200466521856468000819 = (3^2 * 7 * 37 * 97 * 307 * 503 * 709 * 1123 * 7211) and 3*2 + 7 + 37 + 97 + 307 + 503 + 709 + 1123 + 7211 = 10000 = 10^4.

Least 12-digit
For consecutive primes 101904986921 and 101904986927, 101904986921^2 + 101904986927^2 = 20769252719961221963570, add 1 to get 20769252719961221963571 = (3 * 7 * 1171 * 53201 * 1997339 * 7948279) and 3 + 7 + 1171 + 53201 + 1997339 + 7948279 = 10000000 = 10^7.

Least 13-digit
For consecutive primes 1004175782717 and 1004175782719, 1004175782717^2 + 1004175782719^2 = 2016738005194615894935050, add 1 to get 2016738005194615894935051 = (3 * 7^2 * 11 * 59 * 337 * 1973 * 23143 * 33721 * 40739) and 3 + 7*2 + 11 + 59 + 337 + 1973 + 23143 + 33721 + 40739 = 100000 = 10^5.

Least 14-digit
For consecutive primes 10018139795273 and 10018139795291, 10018139795273^2 + 10018139795291^2 = 200726249915625745738919210, add 1 to get 200726249915625745738919211 = (3^2 * 37 * 71 * 439 * 1429 * 4507 * 5261 * 7027 * 81223) and 3*2 + 37 + 71 + 439 + 1429 + 4507 + 5261 + 7027 + 81223 = 100000 = 10^5.

Solutions in ascending order of p :

I found 94 solutions with p < 5*10^10.

1 2-digit solution.
1 5-digit solution.
1 7-digit solution.
4 8-digit solutions.
9 9-digit solutions.
28 10-digit solutions.
50 11-digit solutions.

11, 56681, 4341083, 44586709, 1031230009, 2921126083 and 37978296617 respectively the least prime of two consecutive primes which give 10, 10^2, 10^3, 10^4 10^5, 10^6, 10^7 and 10^8 as result.

Let s = 10^e be the sum of the prime factors with multiplicity of x = p^2 + q^2 + 1 with p, q consecutive primes.

p, q, x, s, e
11 13 291 100 2
56681 56687 6426151731 1000 3
4341083 4341107 37690211598339 10000 4
41512687 41512729 3446609850987411 10000 4
44586709 44586761 3975953875921803 100000 5
66103351 66103363 8739307613338971 10000 4
91416967 91416991 16714128098973171 10000 4
138018481 138018499 38098207163760363 100000 5
168218471 168218509 56594920755760923 100000 5
189632411 189632413 71920903361875491 10000 4
190889693 190889731 72877764294886611 100000 5
427396637 427396663 365335792862445339 100000 5
536753377 536753389 576208388323889451 100000 5
678782777 678782801 921492149285237331 100000 5
738896399 738896429 1091935821244119243 10000 4
779714669 779714681 1215909948820711323 100000 5
1031230009 1031230027 2126870700048560811 1000000 6
1467190433 1467190463 4305295621404881859 100000 5
1472196983 1472197031 4334728054839517251 100000 5
1582425661 1582425673 5008141983160789851 100000 5
1664014223 1664014241 5537886728597099811 10000 4
2068586831 2068586851 8558103037516718763 10000 4
2594494229 2594494291 13462800930345897123 10000 4
2755223071 2755223077 15182508375004018971 1000000 6
2887935887 2887935917 16680347548121107659 100000 5
2921126083 2921126107 17065955325779898339 10000000 7
3212345593 3212345651 20638328790365135451 100000 5
3671937187 3671937209 26966245572112182651 100000 5
3706260721 3706260737 27472737182655223011 10000 4
3803892181 3803892221 28939191753657249603 100000 5
4391730617 4391730641 38574595835393471571 1000000 6
4570439537 4570439557 41777835305563130619 1000000 6
4597861451 4597861453 42280659863575096611 100000 5
4631139811 4631139823 42894912009205587051 100000 5
4700866271 4700866277 44196287452061286171 1000000 6
4810374929 4810374977 46279414376899505571 10000 4
6089297579 6089297597 74159090230445235651 100000 5
6637461631 6637461637 88111793885643899931 100000 5
6793611469 6793611493 92306313909550027011 1000000 6
7789220711 7789220717 121343918662813339611 10000 4
8088015409 8088015443 130831987062427923531 10000 4
8365335107 8365335127 139957663239426807579 1000000 6
9378919721 9378919723 175928270303441114571 10000000 7
9406192457 9406192507 176952914016866641899 100000 5
10011656233 10011656273 200466521856468000819 10000 4
10934349319 10934349331 239119990322255911323 100000 5
11051204993 11051204999 244258263727230720051 100000 5
11116465229 11116465247 247151598775324793451 10000 4
11931798293 11931798331 284735621916492099411 10000000 7
12196153757 12196153763 297492333075124275219 1000000 6
13243461557 13243461563 350778548182395267219 100000 5
13744743731 13744743737 377835960626664525531 100000 5
13905270721 13905270749 386713108427294880843 10000000 7
14361547393 14361547411 412508087559785899371 1000000 6
15004764521 15004764563 450285917921720940411 100000 5
15283942651 15283942699 467197807385372312403 100000 5
16050329681 16050329693 515226166122787036011 10000 4
17336768279 17336768299 601127069412951975243 10000 4
17455666673 17455666687 609400598486524444899 100000 5
17660397169 17660397199 623779257393270260163 10000 4
18690564071 18690564073 698674370659074442371 1000000 6
20231403317 20231403331 818619360916716498051 10000 4
21739039423 21739039429 945171670330364818971 10000 4
21940293251 21940293253 962752935967513471011 100000 5
25324731329 25324731379 1282684036304541347883 100000 5
25648215233 25648215343 1315661894919193851939 10000 4
25826133041 25826133061 1333978296735909137403 1000000 6
26643232703 26643232723 1419723698798146680939 100000 5
27071331893 27071331943 1465714023629051118699 100000 5
27484037701 27484037741 1510744658898701748483 1000000 6
27537430187 27537430213 1516620124039744080339 100000 5
29476153753 29476153829 1737687284621367346251 10000 4
30100732141 30100732163 1812108152172893102451 100000 5
31596633941 31596633947 1996694553182265990291 10000 4
32426120159 32426120173 2102906538039743735211 100000 5
33400513919 33400513967 2231188663313074815651 10000000 7
33716440073 33716440097 2273596664010789614739 100000 5
34085713891 34085713897 2323671783327249286491 10000 4
34477491949 34477491967 2377394903427909347691 100000 5
34764105329 34764105347 2417086039903120188651 100000 5
35155518007 35155518091 2471820898587129536331 1000000 6
37052788961 37052788999 2745818342392818881523 1000000 6
37978296617 37978296619 2884702028009580475851 100000000 8
38163798859 38163798917 2912951091127420074771 10000 4
38521727681 38521727747 2967847012143111333771 10000 4
38904769633 38904769693 3027162205062170268939 10000 4
40304706511 40304706527 3248938735165442194851 1000000 6
40862549411 40862549417 3339495889223183486811 100000 5
41047985327 41047985351 3369874200781517890131 10000000 7
42790818173 42790818181 3662108240514147206691 100000 5
43807079627 43807079633 3838120451418321873819 10000 4
46345549807 46345549823 4295819975309293068579 1000000 6
47850448553 47850448577 4579330855743421118739 10000 4
48006342719 48006342779 4609217888268929755803 100000 5

2-digit solutions

For consecutive primes 11 and 13, 11^2 + 13^2 = 290, add 1 to get 291 = (3 * 97) and 3 + 97 = 100 = 10^2.

5-digit solutions

For consecutive primes 56681 and 56687, 56681^2 + 56687^2 = 6426151730, add 1 to get 6426151731 = (3^2 * 29 * 179 * 263 * 523) and 32 + 29 + 179 + 263 + 523 = 1000 = 10^3.

7-digit solutions

For consecutive primes 4341083 and 4341107, 4341083^2 + 4341107^2 = 37690211598338, add 1 to get 37690211598339 = (3 * 19 * 67 * 467 * 3643 * 5801) and 3 + 19 + 67 + 467 + 3643 + 5801 = 10000 = 10^4.

8-digit solutions

For consecutive primes 41512687 and 41512729, 41512687^2 + 41512729^2 = 3446609850987410, add 1 to get 3446609850987411 = (3 * 79 * 719 * 1571 * 2521 * 5107) and 3 + 79 + 719 + 1571 + 2521 + 5107 = 10000 = 10^4.

For consecutive primes 44586709 and 44586761, 44586709^2 + 44586761^2 = 3975953875921802, add 1 to get 3975953875921803 = (3 * 29 * 61 * 1733 * 4621 * 93553) and 3 + 29 + 61 + 1733 + 4621 + 93553 = 100000 = 10^5.

For consecutive primes 66103351 and 66103363, 66103351^2 + 66103363^2 = 8739307613338970, add 1 to get 8739307613338971 = (3 * 101 * 1319 * 2207 * 2699 * 3671) and 3 + 101 + 1319 + 2207 + 2699 + 3671 = 10000 = 10^4.

For consecutive primes 91416967 and 91416991, 91416967^2 + 91416991^2 = 16714128098973170, add 1 to get 16714128098973171 = (3 * 419 * 499 * 2371 * 3251 * 3457) and 3 + 419 + 499 + 2371 + 3251 + 3457 = 10000 = 10^4.

9-digit solutions

For consecutive primes 138018481 and 138018499, 138018481^2 + 138018499^2 = 38098207163760362, add 1 to get 38098207163760363 = (3^2 * 139 * 18541 * 37423 * 43891) and 3*2 + 139 + 18541 + 37423 + 43891 = 100000 = 10^5.

For consecutive primes 168218471 and 168218509, 168218471^2 + 168218509^2 = 56594920755760922, add 1 to get 56594920755760923 = (3 * 59 * 571 * 613 * 10331 * 88423) and 3 + 59 + 571 + 613 + 10331 + 88423 = 100000 = 10^5.

For consecutive primes 189632411 and 189632413, 189632411^2 + 189632413^2 = 71920903361875490, add 1 to get 71920903361875491 = (3 * 7^3 * 11 * 31 * 73 * 419 * 773 * 8669) and 3 + 7*3 + 11 + 31 + 73 + 419 + 773 + 8669 = 10000 = 10^4.

For consecutive primes 190889693 and 190889731, 190889693^2 + 190889731^2 = 72877764294886610, add 1 to get 72877764294886611 = (3 * 59 * 263 * 1483 * 12289 * 85903) and 3 + 59 + 263 + 1483 + 12289 + 85903 = 100000 = 10^5.

For consecutive primes 427396637 and 427396663, 427396637^2 + 427396663^2 = 365335792862445338, add 1 to get 365335792862445339 = (3 * 11 * 317 * 25463 * 34847 * 39359) and 3 + 11 + 317 + 25463 + 34847 + 39359 = 100000 = 10^5.

For consecutive primes 536753377 and 536753389, 536753377^2 + 536753389^2 = 576208388323889450, add 1 to get 576208388323889451 = (3^5 * 13 * 137 * 1459 * 10369 * 88007) and 3*5 + 13 + 137 + 1459 + 10369 + 88007 = 100000 = 10^5.

For consecutive primes 678782777 and 678782801, 678782777^2 + 678782801^2 = 921492149285237330, add 1 to get 921492149285237331 = (3 * 41 * 827 * 5419 * 23971 * 69739) and 3 + 41 + 827 + 5419 + 23971 + 69739 = 100000 = 10^5.

For consecutive primes 738896399 and 738896429, 738896399^2 + 738896429^2 = 1091935821244119242, add 1 to get 1091935821244119243 = (3 * 7^4 * 229^2 * 503 * 691 * 8317) and 3 + 7*4 + 229*2 + 503 + 691 + 8317 = 10000 = 10^4.

For consecutive primes 779714669 and 779714681, 779714669^2 + 779714681^2 = 1215909948820711322, add 1 to get 1215909948820711323 = (3 * 53 * 3331 * 4339 * 6143 * 86131) and 3 + 53 + 3331 + 4339 + 6143 + 86131 = 100000 = 10^5.

10-digit solutions

For consecutive primes 1031230009 and 1031230027, 1031230009^2 + 1031230027^2 = 2126870700048560810, add 1 to get 2126870700048560811 = (3 * 19 * 317 * 3539 * 34591 * 961531) and 3 + 19 + 317 + 3539 + 34591 + 961531 = 1000000 = 10^6.

For consecutive primes 1467190433 and 1467190463, 1467190433^2 + 1467190463^2 = 4305295621404881858, add 1 to get 4305295621404881859 = (3 * 7 * 7243 * 25321 * 29387 * 38039) and 3 + 7 + 7243 + 25321 + 29387 + 38039 = 100000 = 10^5.

For consecutive primes 1472196983 and 1472197031, 1472196983^2 + 1472197031^2 = 4334728054839517250, add 1 to get 4334728054839517251 = (3^3 * 7^2 * 67 * 131 * 281 * 15889 * 83609) and 3*3 + 7*2 + 67 + 131 + 281 + 15889 + 83609 = 100000 = 10^5.

For consecutive primes 1582425661 and 1582425673, 1582425661^2 + 1582425673^2 = 5008141983160789850, add 1 to get 5008141983160789851 = (3 * 7 * 9743 * 23627 * 24733 * 41887) and 3 + 7 + 9743 + 23627 + 24733 + 41887 = 100000 = 10^5.

For consecutive primes 1664014223 and 1664014241, 1664014223^2 + 1664014241^2 = 5537886728597099810, add 1 to get 5537886728597099811 = (3^2 * 11 * 37 * 47 * 113 * 151 * 167 * 1399 * 8069) and 3*2 + 11 + 37 + 47 + 113 + 151 + 167 + 1399 + 8069 = 10000 = 10^4.

For consecutive primes 2068586831 and 2068586851, 2068586831^2 + 2068586851^2 = 8558103037516718762, add 1 to get 8558103037516718763 = (3 * 7 * 11 * 23^2 * 41 * 73 * 1669 * 2467 * 5683) and 3 + 7 + 11 + 23*2 + 41 + 73 + 1669 + 2467 + 5683 = 10000 = 10^4.

For consecutive primes 2594494229 and 2594494291, 2594494229^2 + 2594494291^2 = 13462800930345897122, add 1 to get 13462800930345897123 = (3 * 7^4 * 109 * 1117 * 1609 * 1783 * 5351) and 3 + 7*4 + 109 + 1117 + 1609 + 1783 + 5351 = 10000 = 10^4.

For consecutive primes 2755223071 and 2755223077, 2755223071^2 + 2755223077^2 = 15182508375004018970, add 1 to get 15182508375004018971 = (3 * 13 * 47 * 40927 * 313543 * 645467) and 3 + 13 + 47 + 40927 + 313543 + 645467 = 1000000 = 10^6.

For consecutive primes 2887935887 and 2887935917, 2887935887^2 + 2887935917^2 = 16680347548121107658, add 1 to get 16680347548121107659 = (3 * 673 * 701 * 6229 * 30637 * 61757) and 3 + 673 + 701 + 6229 + 30637 + 61757 = 100000 = 10^5.

For consecutive primes 2921126083 and 2921126107, 2921126083^2 + 2921126107^2 = 17065955325779898338, add 1 to get 17065955325779898339 = (3 * 19 * 97 * 443 * 753587 * 9245851) and 3 + 19 + 97 + 443 + 753587 + 9245851 = 10000000 = 10^7.

For consecutive primes 3212345593 and 3212345651, 3212345593^2 + 3212345651^2 = 20638328790365135450, add 1 to get 20638328790365135451 = (3^3 * 7^2 * 41 * 79 * 2081 * 40189 * 57587) and 3*3 + 7*2 + 41 + 79 + 2081 + 40189 + 57587 = 100000 = 10^5.

For consecutive primes 3671937187 and 3671937209, 3671937187^2 + 3671937209^2 = 26966245572112182650, add 1 to get 26966245572112182651 = (3^5 * 7 * 1259 * 8429 * 21817 * 68473) and 3*5 + 7 + 1259 + 8429 + 21817 + 68473 = 100000 = 10^5.

For consecutive primes 3706260721 and 3706260737, 3706260721^2 + 3706260737^2 = 27472737182655223010, add 1 to get 27472737182655223011 = (3 * 7^2 * 23^2 * 61 * 199 * 2341 * 2657 * 4679) and 3 + 7*2 + 23*2 + 61 + 199 + 2341 + 2657 + 4679 = 10000 = 10^4.

For consecutive primes 3803892181 and 3803892221, 3803892181^2 + 3803892221^2 = 28939191753657249602, add 1 to get 28939191753657249603 = (3^2 * 7^2 * 11 * 17 * 449 * 467 * 21617 * 77419) and 3*2 + 7*2 + 11 + 17 + 449 + 467 + 21617 + 77419 = 100000 = 10^5.

For consecutive primes 4391730617 and 4391730641, 4391730617^2 + 4391730641^2 = 38574595835393471570, add 1 to get 38574595835393471571 = (3 * 19 * 281 * 9851 * 473227 * 516619) and 3 + 19 + 281 + 9851 + 473227 + 516619 = 1000000 = 10^6.

For consecutive primes 4570439537 and 4570439557, 4570439537^2 + 4570439557^2 = 41777835305563130618, add 1 to get 41777835305563130619 = (3 * 283 * 317 * 719 * 316471 * 682207) and 3 + 283 + 317 + 719 + 316471 + 682207 = 1000000 = 10^6.

For consecutive primes 4597861451 and 4597861453, 4597861451^2 + 4597861453^2 = 42280659863575096610, add 1 to get 42280659863575096611 = (3 * 1619 * 1879 * 3391 * 18251 * 74857) and 3 + 1619 + 1879 + 3391 + 18251 + 74857 = 100000 = 10^5.

For consecutive primes 4631139811 and 4631139823, 4631139811^2 + 4631139823^2 = 42894912009205587050, add 1 to get 42894912009205587051 = (3^3 * 7^3 * 2081 * 4339 * 5849 * 87701) and 3*3 + 7*3 + 2081 + 4339 + 5849 + 87701 = 100000 = 10^5.

For consecutive primes 4700866271 and 4700866277, 4700866271^2 + 4700866277^2 = 44196287452061286170, add 1 to get 44196287452061286171 = (3 * 7 * 2063 * 11743 * 97787 * 888397) and 3 + 7 + 2063 + 11743 + 97787 + 888397 = 1000000 = 10^6.

For consecutive primes 4810374929 and 4810374977, 4810374929^2 + 4810374977^2 = 46279414376899505570, add 1 to get 46279414376899505571 = (3 * 7 * 11 * 43^2 * 101 * 139 * 359 * 4337 * 4957) and 3 + 7 + 11 + 43*2 + 101 + 139 + 359 + 4337 + 4957 = 10000 = 10^4.

For consecutive primes 6089297579 and 6089297597, 6089297579^2 + 6089297597^2 = 74159090230445235650, add 1 to get 74159090230445235651 = (3 * 1907 * 2131 * 6569 * 11959 * 77431) and 3 + 1907 + 2131 + 6569 + 11959 + 77431 = 100000 = 10^5.

For consecutive primes 6637461631 and 6637461637, 6637461631^2 + 6637461637^2 = 88111793885643899930, add 1 to get 88111793885643899931 = (3 * 7 * 19 * 29^2 * 59 * 107 * 379 * 1117 * 98251) and 3 + 7 + 19 + 29*2 + 59 + 107 + 379 + 1117 + 98251 = 100000 = 10^5.

For consecutive primes 6793611469 and 6793611493, 6793611469^2 + 6793611493^2 = 92306313909550027010, add 1 to get 92306313909550027011 = (3 * 443 * 547 * 1297 * 110323 * 887387) and 3 + 443 + 547 + 1297 + 110323 + 887387 = 1000000 = 10^6.

For consecutive primes 7789220711 and 7789220717, 7789220711^2 + 7789220717^2 = 121343918662813339610, add 1 to get 121343918662813339611 = (3^3 * 17^2 * 809 * 1021 * 2029 * 2917 * 3181) and 3*3 + 17*2 + 809 + 1021 + 2029 + 2917 + 3181 = 10000 = 10^4.

For consecutive primes 8088015409 and 8088015443, 8088015409^2 + 8088015443^2 = 130831987062427923530, add 1 to get 130831987062427923531 = (3 * 7^2 * 17 * 83 * 113 * 677 * 1063 * 1123 * 6907) and 3 + 7*2 + 17 + 83 + 113 + 677 + 1063 + 1123 + 6907 = 10000 = 10^4.

For consecutive primes 8365335107 and 8365335127, 8365335107^2 + 8365335127^2 = 139957663239426807578, add 1 to get 139957663239426807579 = (3 * 7 * 271 * 130523 * 413197 * 455999) and 3 + 7 + 271 + 130523 + 413197 + 455999 = 1000000 = 10^6.

For consecutive primes 9378919721 and 9378919723, 9378919721^2 + 9378919723^2 = 175928270303441114570, add 1 to get 175928270303441114571 = (3 * 7 * 83 * 4327 * 3712757 * 6282823) and 3 + 7 + 83 + 4327 + 3712757 + 6282823 = 10000000 = 10^7.

For consecutive primes 9406192457 and 9406192507, 9406192457^2 + 9406192507^2 = 176952914016866641898, add 1 to get 176952914016866641899 = (3^2 * 7^2 * 43 * 47 * 197 * 571 * 23269 * 75853) and 3*2 + 7*2 + 43 + 47 + 197 + 571 + 23269 + 75853 = 100000 = 10^5.

11-digit solutions

For consecutive primes 10011656233 and 10011656273, 10011656233^2 + 10011656273^2 = 200466521856468000818, add 1 to get 200466521856468000819 = (3^2 * 7 * 37 * 97 * 307 * 503 * 709 * 1123 * 7211) and 3*2 + 7 + 37 + 97 + 307 + 503 + 709 + 1123 + 7211 = 10000 = 10^4.

For consecutive primes 10934349319 and 10934349331, 10934349319^2 + 10934349331^2 = 239119990322255911322, add 1 to get 239119990322255911323 = (3^4 * 7 * 13 * 3181 * 4861 * 42139 * 49787) and 3*4 + 7 + 13 + 3181 + 4861 + 42139 + 49787 = 100000 = 10^5.

For consecutive primes 11051204993 and 11051204999, 11051204993^2 + 11051204999^2 = 244258263727230720050, add 1 to get 244258263727230720051 = (3^2 * 7^2 * 137 * 173 * 211 * 331 * 3499 * 95629) and 3*2 + 7*2 + 137 + 173 + 211 + 331 + 3499 + 95629 = 100000 = 10^5.

For consecutive primes 11116465229 and 11116465247, 11116465229^2 + 11116465247^2 = 247151598775324793450, add 1 to get 247151598775324793451 = (3^2 * 11 * 71 * 101 * 181^2 * 1063 * 1439 * 6947) and 3*2 + 11 + 71 + 101 + 181*2 + 1063 + 1439 + 6947 = 10000 = 10^4.

For consecutive primes 11931798293 and 11931798331, 11931798293^2 + 11931798331^2 = 284735621916492099410, add 1 to get 284735621916492099411 = (3 * 19 * 23 * 8779 * 4530593 * 5460583) and 3 + 19 + 23 + 8779 + 4530593 + 5460583 = 10000000 = 10^7.

For consecutive primes 12196153757 and 12196153763, 12196153757^2 + 12196153763^2 = 297492333075124275218, add 1 to get 297492333075124275219 = (3 * 23 * 2417 * 13487 * 160619 * 823451) and 3 + 23 + 2417 + 13487 + 160619 + 823451 = 1000000 = 10^6.

For consecutive primes 13243461557 and 13243461563, 13243461557^2 + 13243461563^2 = 350778548182395267218, add 1 to get 350778548182395267219 = (3 * 1459 * 3847 * 14407 * 27281 * 53003) and 3 + 1459 + 3847 + 14407 + 27281 + 53003 = 100000 = 10^5.

For consecutive primes 13744743731 and 13744743737, 13744743731^2 + 13744743737^2 = 377835960626664525530, add 1 to get 377835960626664525531 = (3^2 * 7^3 * 13 * 1601 * 2659 * 39019 * 56681) and 3*2 + 7*3 + 13 + 1601 + 2659 + 39019 + 56681 = 100000 = 10^5.

For consecutive primes 13905270721 and 13905270749, 13905270721^2 + 13905270749^2 = 386713108427294880842, add 1 to get 386713108427294880843 = (3 * 83 * 599 * 9371 * 27773 * 9962171) and 3 + 83 + 599 + 9371 + 27773 + 9962171 = 10000000 = 10^7.

For consecutive primes 14361547393 and 14361547411, 14361547393^2 + 14361547411^2 = 412508087559785899370, add 1 to get 412508087559785899371 = (3 * 19 * 1051 * 47087 * 192611 * 759229) and 3 + 19 + 1051 + 47087 + 192611 + 759229 = 1000000 = 10^6.

For consecutive primes 15004764521 and 15004764563, 15004764521^2 + 15004764563^2 = 450285917921720940410, add 1 to get 450285917921720940411 = (3^3 * 11^2 * 53 * 1973 * 1999 * 7451 * 88493) and 3*3 + 11*2 + 53 + 1973 + 1999 + 7451 + 88493 = 100000 = 10^5.

For consecutive primes 15283942651 and 15283942699, 15283942651^2 + 15283942699^2 = 467197807385372312402, add 1 to get 467197807385372312403 = (3^3 * 7 * 11 * 97 * 173 * 7573 * 27259 * 64871) and 3*3 + 7 + 11 + 97 + 173 + 7573 + 27259 + 64871 = 100000 = 10^5.

For consecutive primes 16050329681 and 16050329693, 16050329681^2 + 16050329693^2 = 515226166122787036010, add 1 to get 515226166122787036011 = (3^3 * 31 * 101 * 419 * 811 * 1637 * 2371 * 4621) and 3*3 + 31 + 101 + 419 + 811 + 1637 + 2371 + 4621 = 10000 = 10^4.

For consecutive primes 17336768279 and 17336768299, 17336768279^2 + 17336768299^2 = 601127069412951975242, add 1 to get 601127069412951975243 = (3 * 7^2 * 19 * 23 * 431 * 1229 * 2053 * 2069 * 4159) and 3 + 7*2 + 19 + 23 + 431 + 1229 + 2053 + 2069 + 4159 = 10000 = 10^4.

For consecutive primes 17455666673 and 17455666687, 17455666673^2 + 17455666687^2 = 609400598486524444898, add 1 to get 609400598486524444899 = (3^2 * 13 * 23 * 29 * 131 * 139 * 739 * 6263 * 92657) and 3*2 + 13 + 23 + 29 + 131 + 139 + 739 + 6263 + 92657 = 100000 = 10^5.

For consecutive primes 17660397169 and 17660397199, 17660397169^2 + 17660397199^2 = 623779257393270260162, add 1 to get 623779257393270260163 = (3^2 * 7 * 23 * 223 * 449 * 653 * 701 * 1447 * 6491) and 3*2 + 7 + 23 + 223 + 449 + 653 + 701 + 1447 + 6491 = 10000 = 10^4.

For consecutive primes 18690564071 and 18690564073, 18690564071^2 + 18690564073^2 = 698674370659074442370, add 1 to get 698674370659074442371 = (3 * 73 * 443 * 30703 * 475619 * 493159) and 3 + 73 + 443 + 30703 + 475619 + 493159 = 1000000 = 10^6.

For consecutive primes 20231403317 and 20231403331, 20231403317^2 + 20231403331^2 = 818619360916716498050, add 1 to get 818619360916716498051 = (3^2 * 11 * 43 * 109 * 257 * 541 * 787 * 3187 * 5059) and 3*2 + 11 + 43 + 109 + 257 + 541 + 787 + 3187 + 5059 = 10000 = 10^4.

For consecutive primes 21739039423 and 21739039429, 21739039423^2 + 21739039429^2 = 945171670330364818970, add 1 to get 945171670330364818971 = (3 * 7^2 * 23 * 137 * 211 * 331 * 2693 * 3259 * 3329) and 3 + 7*2 + 23 + 137 + 211 + 331 + 2693 + 3259 + 3329 = 10000 = 10^4.

For consecutive primes 21940293251 and 21940293253, 21940293251^2 + 21940293253^2 = 962752935967513471010, add 1 to get 962752935967513471011 = (3 * 7^2 * 11^2 * 103 * 389 * 1327 * 11789 * 86353) and 3 + 7*2 + 11*2 + 103 + 389 + 1327 + 11789 + 86353 = 100000 = 10^5.

For consecutive primes 25324731329 and 25324731379, 25324731329^2 + 25324731379^2 = 1282684036304541347882, add 1 to get 1282684036304541347883 = (3^3 * 7^2 * 71 * 1013 * 11633 * 16339 * 70921) and 3*3 + 7*2 + 71 + 1013 + 11633 + 16339 + 70921 = 100000 = 10^5.

For consecutive primes 25648215233 and 25648215343, 25648215233^2 + 25648215343^2 = 1315661894919193851938, add 1 to get 1315661894919193851939 = (3 * 7 * 13 * 37 * 83 * 311 * 439 * 683 * 3257 * 5167) and 3 + 7 + 13 + 37 + 83 + 311 + 439 + 683 + 3257 + 5167 = 10000 = 10^4.

For consecutive primes 25826133041 and 25826133061, 25826133041^2 + 25826133061^2 = 1333978296735909137402, add 1 to get 1333978296735909137403 = (3 * 431 * 2371 * 5333 * 90523 * 901339) and 3 + 431 + 2371 + 5333 + 90523 + 901339 = 1000000 = 10^6.

For consecutive primes 26643232703 and 26643232723, 26643232703^2 + 26643232723^2 = 1419723698798146680938, add 1 to get 1419723698798146680939 = (3 * 7 * 11 * 13 * 23 * 359 * 457 * 1063 * 1217 * 96847) and 3 + 7 + 11 + 13 + 23 + 359 + 457 + 1063 + 1217 + 96847 = 100000 = 10^5.

For consecutive primes 27071331893 and 27071331943, 27071331893^2 + 27071331943^2 = 1465714023629051118698, add 1 to get 1465714023629051118699 = (3^3 * 7 * 31 * 269 * 509 * 829 * 34543 * 63803) and 3*3 + 7 + 31 + 269 + 509 + 829 + 34543 + 63803 = 100000 = 10^5.

For consecutive primes 27484037701 and 27484037741, 27484037701^2 + 27484037741^2 = 1510744658898701748482, add 1 to get 1510744658898701748483 = (3^2 * 13^2 * 17 * 23 * 97 * 401 * 70289 * 929141) and 3*2 + 13*2 + 17 + 23 + 97 + 401 + 70289 + 929141 = 1000000 = 10^6.

For consecutive primes 27537430187 and 27537430213, 27537430187^2 + 27537430213^2 = 1516620124039744080338, add 1 to get 1516620124039744080339 = (3 * 4211 * 9551 * 11681 * 19571 * 54983) and 3 + 4211 + 9551 + 11681 + 19571 + 54983 = 100000 = 10^5.

For consecutive primes 29476153753 and 29476153829, 29476153753^2 + 29476153829^2 = 1737687284621367346250, add 1 to get 1737687284621367346251 = (3^3 * 29 * 547 * 613 * 709 * 1283 * 1327 * 5483) and 3*3 + 29 + 547 + 613 + 709 + 1283 + 1327 + 5483 = 10000 = 10^4.

For consecutive primes 30100732141 and 30100732163, 30100732141^2 + 30100732163^2 = 1812108152172893102450, add 1 to get 1812108152172893102451 = (3^2 * 7 * 11^2 * 29 * 557 * 8599 * 26713 * 64067) and 3*2 + 7 + 11*2 + 29 + 557 + 8599 + 26713 + 64067 = 100000 = 10^5.

For consecutive primes 31596633941 and 31596633947, 31596633941^2 + 31596633947^2 = 1996694553182265990290, add 1 to get 1996694553182265990291 = (3 * 7^2 * 23 * 53 * 509 * 1171 * 1831 * 3067 * 3329) and 3 + 7*2 + 23 + 53 + 509 + 1171 + 1831 + 3067 + 3329 = 10000 = 10^4.

For consecutive primes 32426120159 and 32426120173, 32426120159^2 + 32426120173^2 = 2102906538039743735210, add 1 to get 2102906538039743735211 = (3^2 * 11 * 23 * 29 * 47 * 107 * 3989 * 21317 * 74471) and 3*2 + 11 + 23 + 29 + 47 + 107 + 3989 + 21317 + 74471 = 100000 = 10^5.

For consecutive primes 33400513919 and 33400513967, 33400513919^2 + 33400513967^2 = 2231188663313074815650, add 1 to get 2231188663313074815651 = (3 * 7 * 31 * 302831 * 1357009 * 8340119) and 3 + 7 + 31 + 302831 + 1357009 + 8340119 = 10000000 = 10^7.

For consecutive primes 33716440073 and 33716440097, 33716440073^2 + 33716440097^2 = 2273596664010789614738, add 1 to get 2273596664010789614739 = (3 * 11^2 * 17^2 * 59 * 227 * 1867 * 9857 * 87931) and 3 + 11*2 + 17*2 + 59 + 227 + 1867 + 9857 + 87931 = 100000 = 10^5.

For consecutive primes 34085713891 and 34085713897, 34085713891^2 + 34085713897^2 = 2323671783327249286490, add 1 to get 2323671783327249286491 = (3^4 * 7^4 * 17 * 29 * 1021 * 1999 * 3361 * 3533) and 3*4 + 7*4 + 17 + 29 + 1021 + 1999 + 3361 + 3533 = 10000 = 10^4.

For consecutive primes 34477491949 and 34477491967, 34477491949^2 + 34477491967^2 = 2377394903427909347690, add 1 to get 2377394903427909347691 = (3^4 * 19 * 181 * 1951 * 5501 * 9613 * 82723) and 3*4 + 19 + 181 + 1951 + 5501 + 9613 + 82723 = 100000 = 10^5.

For consecutive primes 34764105329 and 34764105347, 34764105329^2 + 34764105347^2 = 2417086039903120188650, add 1 to get 2417086039903120188651 = (3 * 2477 * 17497 * 22453 * 27997 * 29573) and 3 + 2477 + 17497 + 22453 + 27997 + 29573 = 100000 = 10^5.

For consecutive primes 35155518007 and 35155518091, 35155518007^2 + 35155518091^2 = 2471820898587129536330, add 1 to get 2471820898587129536331 = (3 * 37 * 1487 * 69403 * 460543 * 468527) and 3 + 37 + 1487 + 69403 + 460543 + 468527 = 1000000 = 10^6.

For consecutive primes 37052788961 and 37052788999, 37052788961^2 + 37052788999^2 = 2745818342392818881522, add 1 to get 2745818342392818881523 = (3 * 163 * 1319 * 26161 * 214811 * 757543) and 3 + 163 + 1319 + 26161 + 214811 + 757543 = 1000000 = 10^6.

For consecutive primes 37978296617 and 37978296619, 37978296617^2 + 37978296619^2 = 2884702028009580475850, add 1 to get 2884702028009580475851 = (3 * 7 * 11 * 4999 * 48704479 * 51290501) and 3 + 7 + 11 + 4999 + 48704479 + 51290501 = 100000000 = 10^8.

For consecutive primes 38163798859 and 38163798917, 38163798859^2 + 38163798917^2 = 2912951091127420074770, add 1 to get 2912951091127420074771 = (3^6 * 13 * 19 * 43 * 103 * 223 * 887 * 3691 * 5003) and 3*6 + 13 + 19 + 43 + 103 + 223 + 887 + 3691 + 5003 = 10000 = 10^4.

For consecutive primes 38521727681 and 38521727747, 38521727681^2 + 38521727747^2 = 2967847012143111333770, add 1 to get 2967847012143111333771 = (3^3 * 23 * 709 * 769 * 839 * 1039 * 2371 * 4241) and 3*3 + 23 + 709 + 769 + 839 + 1039 + 2371 + 4241 = 10000 = 10^4.

For consecutive primes 38904769633 and 38904769693, 38904769633^2 + 38904769693^2 = 3027162205062170268938, add 1 to get 3027162205062170268939 = (3^3 * 79 * 109 * 643 * 1367 * 1493 * 3119 * 3181) and 3*3 + 79 + 109 + 643 + 1367 + 1493 + 3119 + 3181 = 10000 = 10^4.

For consecutive primes 40304706511 and 40304706527, 40304706511^2 + 40304706527^2 = 3248938735165442194850, add 1 to get 3248938735165442194851 = (3 * 47 * 2383 * 53993 * 263227 * 680347) and 3 + 47 + 2383 + 53993 + 263227 + 680347 = 1000000 = 10^6.

For consecutive primes 40862549411 and 40862549417, 40862549411^2 + 40862549417^2 = 3339495889223183486810, add 1 to get 3339495889223183486811 = (3^3 * 7 * 23 * 67 * 2153 * 7129 * 9173 * 81439) and 3*3 + 7 + 23 + 67 + 2153 + 7129 + 9173 + 81439 = 100000 = 10^5.

For consecutive primes 41047985327 and 41047985351, 41047985327^2 + 41047985351^2 = 3369874200781517890130, add 1 to get 3369874200781517890131 = (3 * 11 * 569 * 13099 * 1641931 * 8344387) and 3 + 11 + 569 + 13099 + 1641931 + 8344387 = 10000000 = 10^7.

For consecutive primes 42790818173 and 42790818181, 42790818173^2 + 42790818181^2 = 3662108240514147206690, add 1 to get 3662108240514147206691 = (3 * 6899 * 10343 * 18443 * 21839 * 42473) and 3 + 6899 + 10343 + 18443 + 21839 + 42473 = 100000 = 10^5.

For consecutive primes 43807079627 and 43807079633, 43807079627^2 + 43807079633^2 = 3838120451418321873818, add 1 to get 3838120451418321873819 = (3^2 * 7 * 19 * 239 * 773 * 1021 * 2083 * 2293 * 3559) and 3*2 + 7 + 19 + 239 + 773 + 1021 + 2083 + 2293 + 3559 = 10000 = 10^4.

For consecutive primes 46345549807 and 46345549823, 46345549807^2 + 46345549823^2 = 4295819975309293068578, add 1 to get 4295819975309293068579 = (3 * 467 * 1087 * 13049 * 329671 * 655723) and 3 + 467 + 1087 + 13049 + 329671 + 655723 = 1000000 = 10^6.

For consecutive primes 47850448553 and 47850448577, 47850448553^2 + 47850448577^2 = 4579330855743421118738, add 1 to get 4579330855743421118739 = (3^2 * 19^2 * 179 * 409 * 1297 * 1619 * 2113 * 4339) and 3*2 + 19*2 + 179 + 409 + 1297 + 1619 + 2113 + 4339 = 10000 = 10^4.

For consecutive primes 48006342719 and 48006342779, 48006342719^2 + 48006342779^2 = 4609217888268929755802, add 1 to get 4609217888268929755803 = (3^2 * 23^3 * 31 * 167 * 8209 * 12541 * 78977) and 3*2 + 23*3 + 31 + 167 + 8209 + 12541 + 78977 = 100000 = 10^5.

For consecutive primes 48517474801 and 48517474877, 48517474801^2 + 48517474877^2 = 4707890729505996154730, add 1 to get 4707890729505996154731 = (3^2 * 7 * 17 * 29 * 59 * 641 * 5351 * 8803 * 85087) and 3*2 + 7 + 17 + 29 + 59 + 641 + 5351 + 8803 + 85087 = 100000 = 10^5.

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Oscar wrote:

I understand puzzle 1119 as follows.
Ler (p,q) be a pair of consecutive primes; compute the sum of two squares x = p^2 + q^2.
Factorize next integer y = x+1; let z be the sum of the prime factors of y, without repetition.

Is z a power of 10?

For the given example (11,13), y is a semiprime. There are no more solutions with such property.
It's easy to check that y is always a composite number, divisible by 3.
p=2, q=3, x=13, y=14=2*7, z=2+7=9, not a solution;
p=3, q=5, x=34, y=35=5*7, z=5+7=12, not a solution;
p>3, q>p, p^2 == q^2 == 1 mod 3 by Fermat's little theorem,
y>35 and y == 1+1+1 == 0 mod 3.

Let (p,q) be a solution with z = 10^k; if y is a semiprime, its prime factors must be 3 and 10^k-3.
Then x = 3*(10^k - 3) - 1 = 3*10^k - 10.
The sum of two squares theorem states that x can be expressed as the sum of two perfect squares a^2 + b^2 if and only, if in the prime factorization of x, every prime of the form (4*m+3) has even multeplicity.
k = 1, x = 20 = 2^2*5, expressible:
a = 2, b = 4, not consecutive primes.
k = 2, x = 290 = 2*5*29, expressible:
a = 1, b = 17, not consecutive primes;
a = 11, b = 13, the given example.
What about k > 2? Without knowing the complete factorization of x, we can check the following necessary constraint:
if 2^e is the largest power of 2 dividing x, so that x = d*2^e with d odd, then d must be congruent to 1 mod 4.
After substituing k = 3+t, with t>=0, we obtain x = (4*(375*10^t - 2) + 3)*2;
hence e = 1 and d == 3 mod 4, so x is not expressible as the sum of two squares.

If we drop the semiprime constraint, several solutions can be found.
I searched for the first pair (p,q) producing a power z=10^k.
k=1, p=59, q=61
y = 3*7^4
k=2, p=11, q=13
y = 3*97
k=3, p=7759, q=7789
y= 3^3*7^2*103*887
k=4, p=4341083, q=4341107
y = 3*19*67*467*3643*5801
k=5, p=4358161, q=4358191
y = 3^3*7*31*197*331*99431
k=6, p=1031230009, q=1031230027
y = 3*19*317*3539*34591*961531

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