Prime numbers related to 37

Sender: chongo@sgi.com
Date: 1997年12月31日 14:08:48 -0800
From: Landon Curt Noll 
Organization: Silicon Graphics
To: tom@magliery.com
Subject: 37
Saw your 37 reference to lavarand ... amusing.
I've been working in computational number theory and in particular
prime number testing:
 http://reality.sgi.com/csp/ioccc/noll/bio.html
for a while now. I have a particular interest in unusual primality
tests.
Anyway I had some spare Cray cpu cycles to burn, so I spent
some time trying to come up with various primes that involve
the number 37. Some of my results are more interesting that
others ...
I have the decimal expansions for all of these numbers. I can
supply them to you should you want them. But to keep this EMail
message small, I'll only include some of the smaller values.
chongo  /\oo/\
=-=
These numbers are prime:
 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.
Now a more interesting are numbers of the form:
 3737..373
These numbers may be expressed as:
 (370*((100^n)-1)/99)+3
for n>= 1. 
We were able to show that these numbers are prime
for n == 1, 10, 13, 40, 157, 424, 946 and 1441.
We searched for primes for n < 2400 (i.e., for numbers up to 4801 digits) on the Cray. The task took about 37 minutes of Cray time (our algorithm to test primality was not as optimal as it could have been ... in addition we slowed it down somewhat to stretch it out to 37 hours :-) ). Using the above search, we know that these numbers 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. That is these primes read the same forwards and backwards! The smallest 37 digit prime is: 1000000000000000000000000000000000067 The largest 37 digit prime is: 9999999999999999999999999999999999919 Here are 37 primes that are 37 digits long each of which contain the digits 37 that were randomly selected by http://lavarand.sgi.com: 8675893740993106923975426099303977519 6641308249070305332690490651668271337 4354721710654452924133214267196084377 2789578537548353120046029366730200267 4059206136617859455910513242375436809 3010515377256653594299862483877204337 1275485597520029132884590333853750303 6284104451963624029892913921775637657 6323205291630559271784919428647273789 3413754401985431348053508771376553633 1914344700948894862792335590726723779 6126601510797836637484025823603668677 6584137847380527873939774425090045713 5085551081537015819528901379786929817 9323910615479407963799946736108657501 9516634802119507183537370742060670183 5740900477236337850512488771327644953 5099518794538533914206863513799582327 5665806412723036937279638553380882921 2633842637589500389983313037082183827 4029937483323469659377094101634086113 3245775377978544991969924793069089909 9643793620398723833140115407659231401 9751497689312364630825445233723669247 2862033198804598348374034564950088897 1467028356867510231101693703999838831 4710917210801776917208483749699874601 9500324536027497480904646238250224137 6235658617212132899845960873753106663 5843064662603893370272505298313121407 9465707317863702602013584832046537973 1031641650253872852643722015344540051 8575546461837029795310544141872846161 5134706743960231406003133786230333711 1303728589342054662830190549777533773 9059280883210744634819194459651843781 7159203767132216923313691898848268759 37!+1 = 13763753091226345046315979581580902400000001 is prime. 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 is 1 prime < 100 that contains the digits 37 (in order). 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. Enjoy.