   ## miscellaneous primes related to 37

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Here are some other primes related to 37:

• 21*2^37-1 is prime = 2886218022911
• 27*2^37-1 is prime = 3710851743743 (has a 37 in it)
• 31*2^37-1 is prime = 4260607557631
• 31*2^837-1 is prime = (254 digits) (has a 37 in it)
• 45*2^37-1 is prime = 6184752906239
• 65*2^376-1 is prime = (116 digits) (has a 37 in it)
• 69*2^37-1 is prime = 9483287789567
• 69*2^377-1 is prime = (116 digits) (has a 37 in it)
• 75*2^237-1 is prime = (74 digits) (has a 37 in it)
• 119*2^376-1 is prime = (116 digits)
• 121*2^37-1 is prime = 16630113370111 (has a 37 in it)
• 121*2^373-1 is prime = (115 digits) (has a 37 in it)
• 133*2^375-1 is prime = (116 digits) (has a 37 in it)
• 137*2^18-1 is prime = 35913727 (has a 37 in it)
• 137*2^38-1 is prime = 37658273251327 (has a 37 in it)
• 137*2^62-1 is prime = 631800984524552142847
• 139*2^37-1 is prime = 19104014532607
• 169*2^137-1 is prime = 29443952644934963366558878031856039800864767
• 197*2^374-1 is prime = (115 digits)

• 2^37-1 is the product of two primes: 223 * 616318177
• 2^37+1 is the product of 3 primes: 3 * 1777 * 25781083

• 37^37+42 is prime = 10555134955777783414078330085995832946127396083370199442559

• 37^37-48 is prime = 10555134955777783414078330085995832946127396083370199442469

• While 37 is prime, there are no multiple concatenation of 37 that is prime. I.e., 3737 is not prime, 373737 is not prime and so on ... The proof of this is simple:
All such numbers are of the form: 37*((100^n)-1)/99 where n is an integer > 0. When n==1, the number is 37*1 which is prime. When n > 1, the number is divisible by both 37 and ((100^n)-1)/99 and therefore is not prime.

These numbers may be expressed as:

(370*((100^n)-1)/99)+3
for n >= 1.

I was able to prove that these numbers are prime for n == 1, 10, 13, 40, 157, 424, 946 and 1441. There are no such for primes for 1441 < n < 2400

In particular, these are prime:

 373 3 digits 373737373737373737373 21 digits 373737373737373737373737373 27 digits 37(repeated 40 times)3 81 digits 37(repeated 147 times)3 314 digits 37(repeated 424 times)3 849 digits 37(repeated 946 times)3 1893 digits 37(repeated 1441 times)3 2883 digits

One should note that these primes are all palindromes in base 10. That is the decimal digits of these primes read the same forwards and backwards!

• These sums of the 37th powers of primes are prime:
 2^37 + 3^37 + 5^37 + 7^37 + ... + 881^37 is prime (110 digits) 2^37 + 3^37 + 5^37 + 7^37 + ... + 2053^37 is prime (124 digits) 2^37 + 3^37 + 5^37 + 7^37 + ... + 2267^37 is prime (125 digits) 2^37 + 3^37 + 5^37 + 7^37 + ... + 5443^37 is prime (140 digits) 2^37 + 3^37 + 5^37 + 7^37 + ... + 8753^37 is prime (148 digits) 2^37 + 3^37 + 5^37 + 7^37 + ... + 9029^37 is prime (148 digits)

• There are 6 primes < 1000 that contain the digits 37

• There are 58 primes < 10000 that contain the digits 37

• There are 525 primes < 100000 that contain the digits 37
• There are 3 primes < 100000 that contain the digits 3737

• There are 4968 primes < 1000000 that contain the digits 37
• There are 36 primes < 1000000 that contain the digits 3737 (darn)

• There are 48695 primes < 10000000 that contain the digits 37
• There are 391 primes < 10000000 that contain the digits 3737
• There are 1 prime < 10000000 that contain the digits 373737 (5373737)

• There are 476235 primes < 100000000 that contain the digits 37
• There are 3760 primes < 100000000 that contain the digits 3737
• There are 27 primes < 100000000 that contain the digits 373737
• There are 0 primes < 100000000 that contain the digits 37373737

The 37 smallest 37 decimal digit primes:

1. 1000000000000000000000000000000000067
2. 1000000000000000000000000000000000123
3. 1000000000000000000000000000000000141
4. 1000000000000000000000000000000000159
5. 1000000000000000000000000000000000163
6. 1000000000000000000000000000000000291
7. 1000000000000000000000000000000000537
8. 1000000000000000000000000000000000561
9. 1000000000000000000000000000000000709
10. 1000000000000000000000000000000000849
11. 1000000000000000000000000000000000963
12. 1000000000000000000000000000000001123
13. 1000000000000000000000000000000001149
14. 1000000000000000000000000000000001167
15. 1000000000000000000000000000000001201
16. 1000000000000000000000000000000001243
17. 1000000000000000000000000000000001263
18. 1000000000000000000000000000000001321
19. 1000000000000000000000000000000001383
20. 1000000000000000000000000000000001471
21. 1000000000000000000000000000000001527
22. 1000000000000000000000000000000001549
23. 1000000000000000000000000000000001563
24. 1000000000000000000000000000000001569
25. 1000000000000000000000000000000001711
26. 1000000000000000000000000000000001719
27. 1000000000000000000000000000000002041
28. 1000000000000000000000000000000002073
29. 1000000000000000000000000000000002139
30. 1000000000000000000000000000000002251
31. 1000000000000000000000000000000002271
32. 1000000000000000000000000000000002473
33. 1000000000000000000000000000000002499
34. 1000000000000000000000000000000002523
35. 1000000000000000000000000000000002719
36. 1000000000000000000000000000000002763
37. 1000000000000000000000000000000002769

The product of the 37 smallest 37 decimal digit primes:

```1000000000000000000000000000000052043000000
0000000000000000000013054973020000000000000
0000000002102467213808200000000000000000024
4285057982545121000000000000002181725970316
7926652110000000000155821600794733974262901
2800000009141922222081799579944229092800044
9126534495024404691492896141813874134923827
5772781939776320609942306050986672550491586
1203012720016465604724521058336867727252831
6105497929582487556380084354722277880737554
2642008553398156681824974919033824768230255
3854426143435841177711964922982920760615347
1583757043795961842294668970569162886436195
7875751155951362259765666517137569955790816
5995586493738586075664652591703642002881653
1093702910138106952914110850943314179550117
5085649651272006859399985921061921690439123
8295755844034842594280705170411548688059920
3548750029390964225240414035684387816495313
2472540826948799026325986362328572813117120
7241615028108273294926974072246605255011891
7366786042407848877773024033727111663383808
1168350255202389857437276783488767262824578
0252882046330599200661386956687456070368524
7494952828340236072636257313456581590453392
4688672273658095631670341877717521225440524
1953344999136256262234830435858428109010317
7115046362850852720888452563759292305607363
9030575690167131414680488438901962462399018
1107323915786951767029668250815947304257390
3153521929020382759899740077561513960267699```

The 37 largest 37 decimal digit primes:

1. 9999999999999999999999999999999999919
2. 9999999999999999999999999999999999877
3. 9999999999999999999999999999999999823
4. 9999999999999999999999999999999999623
5. 9999999999999999999999999999999999611
6. 9999999999999999999999999999999999571
7. 9999999999999999999999999999999999557
8. 9999999999999999999999999999999999511
9. 9999999999999999999999999999999999371
10. 9999999999999999999999999999999999247
11. 9999999999999999999999999999999999239
12. 9999999999999999999999999999999999113
13. 9999999999999999999999999999999999091
14. 9999999999999999999999999999999998987
15. 9999999999999999999999999999999998951
16. 9999999999999999999999999999999998939
17. 9999999999999999999999999999999998839
18. 9999999999999999999999999999999998819
19. 9999999999999999999999999999999998801
20. 9999999999999999999999999999999998767
21. 9999999999999999999999999999999998689
22. 9999999999999999999999999999999998531
23. 9999999999999999999999999999999998381
24. 9999999999999999999999999999999998347
25. 9999999999999999999999999999999998323
26. 9999999999999999999999999999999998321
27. 9999999999999999999999999999999998179
28. 9999999999999999999999999999999998173
29. 9999999999999999999999999999999998161
30. 9999999999999999999999999999999998057
31. 9999999999999999999999999999999998047
32. 9999999999999999999999999999999998011
33. 9999999999999999999999999999999997979
34. 9999999999999999999999999999999997969
35. 9999999999999999999999999999999997853
36. 9999999999999999999999999999999997831
37. 9999999999999999999999999999999997777

The product of the 37 smallest 37 decimal digit primes:

```9999999999999999999999999999999954285
0000000000000000000000000001008939633
9999999999999999999999985680483697986
0000000000000000000146874623028978084
9999999999999998839971023348708496009
0000000000007340308094064421529719407
9999999961772392330594233672053649264
0000167046460299771663263586537284691
9378671308460847838841443465463542730
8703223019049354622714211625596525424
5414517399780263689001484645164283126
3025543985127690494906338387446372958
2363896128227860954052185090815485012
5641867462686807167797889487794516630
5309251102969125042013978282379404812
8782211164012766503637453004544324430
4111541030793167014391418053527953429
2169975207893961743999202732785868223
4207984989610378972866687693381587362
7232460385918125312572796495688527390
1943313659137322490989869686144693267
4772139663291530628971658708163145506
9419241851896523512150377935479809920
3505639823970092769558411393381218500
5290966668031471823265818887212063463
8422418703131483902737427654212344887
5392494893435447413153975846067876060
8894290757219550849541965963583485799
5006718701985845680181568631518616292
6363423040567174851929283587694172714
6709563032179948735568863234666930777
8073374635062272938557005831396378747
0133846014027112917544259870018760169
2918591155756293784159177905966254895
0232303952653788695745438633031727071
6609956572542266737767744387526551596
5228047642325909510954201280969757457```

And of course, the above product is 37*37 decimal digits long.

For some more 37 fun, you might want to visit the 37 heaven page. [back]

 © 1994-2013 Landon Curt Noll chongo (was here) /\oo/\ \$Revision: 7.3 \$ \$Date: 2012/05/14 09:18:27 \$