## [PrimeNumbers] Re: A small observation

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• Robert Smith wrote to primenumbers@yahoogroups.com ... Expanding by the binomial theorem, (p+1)q-1 is divisible by p. If you make use
Message 1 of 3 , Dec 23, 2006
Robert Smith <rw.smith@...> wrote to

>For prime p, there are no prime factors of (p+1)^q-1 smaller than p
>for all q<p, q prime.
>

Expanding by the binomial theorem, (p+1)q-1 is divisible by p.
If you make use of the fact that all other prime factors of (p+1)^q-1
are of the form
f = k*q+1,
where k must be even if f is odd,
you should readily be able to prove why your observed patterns occur.

Mike

-----Original Message-----
From: rw.smith@...
Sent: Sat, 23 Dec 2006 8.12AM
Subject: [PrimeNumbers] Re: A small observation

A further bit of research shows that for primes q up to 127, certain
values of q (17, 37) do not have prime factors smaller than p for all
p+1 in (p+1)^q-1

The factors of (p+1)^q-1 is given for q=17 for all p to p=127 below.
The smallest factor in each case is p:

p+1 factors of (p+1)^q-1, q=17
3 2*1871*34511
4 3*43691*131071
6 5*239*409*1123*30839
8 7*103*2143*11119*131071
12 11*2693651*74876782031
14 13*103*22771730193675277
18 17^2*7563707819165039903
20 19*689852631578947368421
24 23*307*120574031*341563234253
30 29*103*409*10570676926829627653
32 31*131071*9520972806333758431
38 37*137*4269211*33193361176106273
42 41*1158007*82935963690420160273
44 43*14519*13908821686834391538539
48 47*C_810966991148301450330945841
54 53*11270423*C_472582712990569339157
60 59*137*57223*3659553116774508114611
62 61*48453916488902607769120106731
68 67*1973*309469*1240559*280036765431547
72 71*103*137*114320887*327876819251626913
74 73*137*258469*23146607407263591330467
80 79*152729*4885801*3819842050386190289
84 83*6218272796370530483675222621221
90 89*71041879*C_263765636099232967211329
98 97*2687*11527*4530671*521104791402957529
102 101*307*16661*87313*310429650430900811021
104 103*C_189116552966629126613043005678761
108 107*14281*17783*C_1361621555332298324123027
110 109*6121*12377*4221271*C_1450000477088161373
114 113*307*38183*C_70031851163818972855046051
128 127*239*20231*131071*131105292137*62983048367

Contrast with q=3, where p is only sometimes the smallest factor:

p+1 factors of (p+1)^q-1, q=3
3 2*13
4 3^2*7
6 5*43
8 7*73
12 11*157
14 13*211
18 7^3*17
20 19*421
24 23*601
30 7^2*19*29
32 7*31*151
38 37*1483
42 13*41*139
44 7*43*283
48 13*47*181
54 53*2971
60 7*59*523
62 61*3907
68 13*19^2*67
72 7*71*751
74 7*13*61*73
80 79*6481
84 37*83*193
90 89*8191
98 31*97*313
102 7*19*79*101
104 67*103*163
108 61*107*193
110 109*12211
114 7*113*1873
128 7^2*127*337

However I have a feeling that this may be a case of "sods law of small
numbers". Trouble is I have no maths software to test these results
further.

Regards

Robert Smith
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