# List of Coprimes

Is there a way to generate a list of parametrized length, say $L$, of integers which are relatively prime among them? I would like to take into account their word-length as well, so for example the list would contain only 32-bit numbers.

• Prime@Range[L] will have that property. It feels as if you had omitted something important from the question ... which would explain why Prime@Range[L] is not what you want ... Mar 27, 2017 at 13:38
• You can also use RandomPrime if you want big numbers. Mar 27, 2017 at 14:07
• I wouldn't like to have only prime numbers, but rather a list of relatively prime numbers. I tried though your command and I get something weird: Prime@Range[2^31, 2^31 + 22] outputs numbers that are larger than $2^{31}+22$. Any idea why is that? Mar 27, 2017 at 22:33

Are you looking for a list of mutually coprime integers? The function CoprimeSet[n,m] finds sets of m mutually coprime integers from the first n>m integers.

CoprimeSet[n_, m_] := Pick[#, CoprimeQ @@@ #] &[Subsets[Range[n], {m}]]

For example,

CoprimeSet[8,5]

{{1, 2, 3, 5, 7}, {1, 3, 4, 5, 7}, {1, 3, 5, 7, 8}}

Alternatively, CoprimeSetRandom[n,m,kmin,kmax] finds sets of m mutually coprime integers from a random selection of n integers between kmin and kmax.

CoprimeRandomSet[n_, m_, kmin_, kmax_] :=
Pick[#, CoprimeQ @@@ #] &[Subsets[RandomInteger[{kmin, kmax}, n], {m}]]

For example,

CoprimeRandomSet[10, 5, 1800, 2000]

{{1803, 1888, 1811, 1961, 1951}, {1803, 1930, 1811, 1961, 1951}}

• Nice effort thanks, although I can't seem to make it work for 32-bit numbers. I tried for example CoprimeRandomSet[50, 22, 2^32, 2^32-1] but I got a message for insufficient memory (16GB + 16GB swap in Ubuntu) Mar 27, 2017 at 22:37
• You have asked for subsets of 22 integers from a list of 50 integers, which is a list of billions of sublists. So memory will always be insufficient for such large cases. Assuming you reduce n and m, you must have kmin less than kmax, whereas your example in the comment has kmax less than kmin. Apr 1, 2017 at 15:13
• Correct, that was a typo. I used CoprimeRandomSet[22,22,2^31,2^32-1] to get just 1 set of 22 coprime numbers but I got no output. Apr 4, 2017 at 12:29