$b^{n}+1$ is prime only if $b\equiv0\pmod2$ and $n$ is a power of $2$

Question

Is there any known proof of the following conjectures:

$b^{n}-1$, $n>1$, is prime only if $b=2$ and $n$ is prime.
$b^{n}+1$, $n>1$, is prime only if $b\equiv0\pmod2$ and $n$ is a power of $2$.

All possible $b^n-1$ with $0<b<15$, $1<n<15$

  1^2-1   0                 2^2-1       3 | PRIME
  3^2-1   8                 4^2-1       15       
  5^2-1   24                6^2-1       35       
  7^2-1   48                8^2-1       63       
  9^2-1   80                10^2-1      99       
 11^2-1   120               12^2-1      143       
 13^2-1   168               14^2-1      195       
  1^3-1   0                 2^3-1       7 | PRIME
  3^3-1   26                4^3-1       63       
  5^3-1   124               6^3-1       215       
  7^3-1   342               8^3-1       511       
  9^3-1   728               10^3-1      999       
 11^3-1   1330              12^3-1      1727       
 13^3-1   2196              14^3-1      2743       
  1^4-1   0                 2^4-1       15       
  3^4-1   80                4^4-1       255       
  5^4-1   624               6^4-1       1295       
  7^4-1   2400              8^4-1       4095       
  9^4-1   6560              10^4-1      9999       
 11^4-1   14640             12^4-1      20735       
 13^4-1   28560             14^4-1      38415       
  1^5-1   0                 2^5-1       31 | PRIME
  3^5-1   242               4^5-1       1023       
  5^5-1   3124              6^5-1       7775       
  7^5-1   16806             8^5-1       32767       
  9^5-1   59048             10^5-1      99999       
 11^5-1   161050            12^5-1      248831       
 13^5-1   371292            14^5-1      537823       
  1^6-1   0                 2^6-1       63       
  3^6-1   728               4^6-1       4095       
  5^6-1   15624             6^6-1       46655       
  7^6-1   117648            8^6-1       262143       
  9^6-1   531440            10^6-1      999999       
 11^6-1   1771560           12^6-1      2985983       
 13^6-1   4826808           14^6-1      7529535       
  1^7-1   0                 2^7-1       127 | PRIME
  3^7-1   2186              4^7-1       16383       
  5^7-1   78124             6^7-1       279935       
  7^7-1   823542            8^7-1       2097151       
  9^7-1   4782968           10^7-1      9999999       
 11^7-1   19487170          12^7-1      35831807       
 13^7-1   62748516          14^7-1      105413503       
  1^8-1   0                 2^8-1       255       
  3^8-1   6560              4^8-1       65535       
  5^8-1   390624            6^8-1       1679615       
  7^8-1   5764800           8^8-1       16777215       
  9^8-1   43046720          10^8-1      99999999       
 11^8-1   214358880         12^8-1      429981695       
 13^8-1   815730720         14^8-1      1475789055       
  1^9-1   0                 2^9-1       511       
  3^9-1   19682             4^9-1       262143       
  5^9-1   1953124           6^9-1       10077695       
  7^9-1   40353606          8^9-1       134217727       
  9^9-1   387420488         10^9-1      999999999       
 11^9-1   2357947690        12^9-1      5159780351       
 13^9-1   10604499372       14^9-1      20661046783       
 1^10-1   0                 2^10-1      1023       
 3^10-1   59048             4^10-1      1048575       
 5^10-1   9765624           6^10-1      60466175       
 7^10-1   282475248         8^10-1      1073741823       
 9^10-1   3486784400        10^10-1     9999999999       
11^10-1   25937424600       12^10-1     61917364223       
13^10-1   137858491848      14^10-1     289254654975       
 1^11-1   0                 2^11-1      2047       
 3^11-1   177146            4^11-1      4194303       
 5^11-1   48828124          6^11-1      362797055       
 7^11-1   1977326742        8^11-1      8589934591       
 9^11-1   31381059608       10^11-1     99999999999       
11^11-1   285311670610      12^11-1     743008370687       
13^11-1   1792160394036     14^11-1     4049565169663       
 1^12-1   0                 2^12-1      4095       
 3^12-1   531440            4^12-1      16777215       
 5^12-1   244140624         6^12-1      2176782335       
 7^12-1   13841287200       8^12-1      68719476735       
 9^12-1   282429536480      10^12-1     999999999999       
11^12-1   3138428376720     12^12-1     8916100448255       
13^12-1   23298085122480    14^12-1     56693912375295       
 1^13-1   0                 2^13-1      8191 | PRIME
 3^13-1   1594322           4^13-1      67108863       
 5^13-1   1220703124        6^13-1      13060694015       
 7^13-1   96889010406       8^13-1      549755813887       
 9^13-1   2541865828328     10^13-1     9999999999999       
11^13-1   34522712143930    12^13-1     106993205379071       
13^13-1   302875106592252   14^13-1     793714773254143       
 1^14-1   0                 2^14-1      16383       
 3^14-1   4782968           4^14-1      268435455       
 5^14-1   6103515624        6^14-1      78364164095       
 7^14-1   678223072848      8^14-1      4398046511103       
 9^14-1   22876792454960    10^14-1     99999999999999       
11^14-1   379749833583240   12^14-1     1283918464548863       
13^14-1   3937376385699288  14^14-1     11112006825558015

I looked up Mersenne primes theorems and other related stuff but i did not find anything that would prove those statements...

Sorry, i just figured out i had one more condition in my computations, n>1... — François Huppé, Sep 01 '18 at 22:16
See Few problems on divisibility - almost have the answers, $a \geq 2$ and $a^n+1$ is a prime number, Two number theory questions: primes of the form $a^m+1$ and $2^{2^k}+5$., If $a^{n}-1$ is prime then $a=2$ and $n$ is prime? — Sil, Sep 01 '18 at 22:58

score 1 · Answer 1 · answered Sep 01 '18 at 22:09

This is false. It is true that if $b^n +1$ is prime then you need the conditions in question. But the other direction is false. Note for example that if $b=6$, and $n=8$ one has that $17|6^8+1$ from some easy modular arithmetic, but $6^8+1$ is much bigger than 17, and so isn't prime.

What is less obvious is that this is also false even if $b$ is a power of 2. Fermat actually conjectured that all numbers of the form $2^{(2^k)}+1$ are prime, which is false for $k=5$. In fact, we now suspect that $2^{(2^k)} +1$ is composite for all $k \geq 5.$ See https://en.wikipedia.org/wiki/Fermat_number

$b^{n}+1$ is prime only if $b\equiv0\pmod2$ and $n$ is a power of $2$

1 Answers1