## Prime numbers related to 37

```Sender: chongo@sgi.com
Date: Wed, 31 Dec 1997 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.

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