TITLE: Primes of some particular form.
QUESTION [5 upvotes]: Given an odd prime $p$, Dirichlet's theorem on arithmetic progressions tells us that there are an infinite number of primes of the form $pk+1$. How might I go about showing that there is a $k < p$ for which $pk+1$ is a prime?

REPLY [2 votes]: Don't expect any proof; works for primes up to 500, but with rather unpredictable $k$
    3    2         7
    5    2        11
    7    4        29
   11    2        23
   13    4        53
   17    6       103
   19   10       191
   23    2        47
   29    2        59
   31   10       311
   37    4       149
   41    2        83
   43    4       173
   47    6       283
   53    2       107
   59   12       709
   61    6       367
   67    4       269
   71    8       569
   73    4       293
   79    4       317
   83    2       167
   89    2       179
   97    4       389
  101    6       607
  103    6       619
  107    6       643
  109   10      1091
  113    2       227
  127    4       509
  131    2       263
  137    6       823
  139    4       557
  149    8      1193
  151    6       907
  157   10      1571
  163    4       653
  167   14      2339
  173    2       347
  179    2       359
  181    6      1087
  191    2       383
  193    4       773
  197   18      3547
  199    4       797
  211   10      2111
  223   12      2677
  227   24      5449
  229   12      2749
  233    2       467
  239    2       479
  241    6      1447
  251    2       503
  257    6      1543
  263    6      1579
  269    8      2153
  271    6      1627
  277    4      1109
  281    2       563
  283    6      1699
  293    2       587
  307    4      1229
  311    6      1867
  313    6      1879
  317   26      8243
  331    6      1987
  337   10      3371
  347    6      2083
  349   10      3491
  353   14      4943
  359    2       719
  367    6      2203
  373    4      1493
  379   12      4549
  383   12      4597
  389   24      9337
  397    6      2383
  401    8      3209
  409    4      1637
  419    2       839
  421   10      4211
  431    2       863
  433    4      1733
  439   10      4391
  443    2       887
  449    8      3593
  457   30     13711
  461    6      2767
  463   12      5557
  467    6      2803
  479    8      3833
  487    4      1949
  491    2       983
  499    4      1997