Generating a large prime, $p$, such that $2^{k}$ divides $p-1$ for some $k<p$

Question

I want to generate a prime $p$ of a certain size $2^{k}$ divides $p-1$ for some $k < p$. Is there any trick that I can use to do that instead of a brute-force search?

Answer 1

The following Java program found one in a fraction of a second:

import java.io.IOException;
import java.math.BigInteger;
import java.util.Random;

public class Primefinder {
  public static void main(String[] args) throws IOException {
    Random r = new Random();
    for( int i=0; i<1000; i++ ) {
        BigInteger bi = new BigInteger(320-32, r);
        bi = bi.shiftLeft(32);
        bi = bi.add(BigInteger.ONE);
        if( bi.isProbablePrime(100) ) {
            System.out.println("found in iteration "+i+": "+bi.toString(16));
            break ;
        }
    }
  }
}

Output from one run:

found in iteration 143: 949f3fe33aa137d2f289064432e30d1f7533d306c2d277f873a6fc5969b0fdaec873455100000001

And for good measure, here are ones of size 80 and 160 (after I filtered out a couple of tries where the first few bits happened to be 0s):

found in iteration 20: d070dd4f6b9f00000001
found in iteration 29: a45583b3b54a5946026be213f0366a9200000001

---

For some less random examples, $13\times 2^{316}+1$ is also prime, as are $29\times 2^{75}+1$ and $315\times 2^{151}+1$.