# Matrices whose entries are prime numbers

Discussion in 'Math Research' started by ss54, Feb 15, 2006.

1. ### ss54Guest

Let M be an n by n matrix whose entries are n^2 prime numbers. Is is
true that this matrix is invertible?

Simone Severini

ss54, Feb 15, 2006

2. ### RobertSzeflerGuest

Take an n x n matrix whose all elements are 2. That certainly is not
invertible! Even assuming distinct primes that's not true (although
obviously holds for 2x2 matrices), take for example

3 5 7
11 13 17
29 43 59

It is composed of primes and singular.

RobertSzefler, Feb 15, 2006

3. ### G. A. EdgarGuest

n=2, yes; n=3, no.

A "prime triple" is the situation (p, p+2, p+6) where they
are all primes. It is conjectured that there are infintely
many prime triples. But in any case, there are at least three
of them. Use them as rows in a 3x3 matrix to get a matrix
with rank 2. For example:

[ 5 7 11]
[17 19 23]
[41 43 47]

has rank 2, and it thus not invertible.

G. A. Edgar, Feb 15, 2006
4. ### Peter L. MontgomeryGuest

No. Consider a matrix all of whose entries are the same prime.
Less trivially,

( 11 13 17 19 )
( 101 103 107 109 )
( 191 193 197 199 )
( 821 823 827 829 )

has rank 2.

Peter L. Montgomery, Feb 15, 2006
5. ### Rob ArthanGuest

Here's a minimal counter-example that just uses the first 9 primes:

3 2 5
11 13 7
19 17 23

"Method": take the rows of the matrix to be triples of primes p, q, and r
obeying a linear dependence relation Ap + Bq+ Cr for some A, B, C. Fiddle
around with small A,B and C until you can get p, q and r really small, (A =
3, B = -2 and C = -1 above).

Regards,

Rob.

Rob Arthan, Feb 15, 2006
6. ### ss54Guest

I think that when I wrote the post, I did not think enough. Of course,
"distinct" was implicit.

I need to modify the question:

The elements of the matrix are consecutive primes.

Specifically the examples I considered are the following:

2 3
5 7
rank: 2

2 3 5
7 11 13
17 19 23
rank: 3

2 3 5 7
11 13 17 19
23 29 31 37
41 43 47 53
rank: 4

2 3 5 7 11
13 17 19 23 29
31 37 41 43 47
53 59 61 67 71
73 79 83 89 97
rank: 5

Probably I should go a bit further and try to find a counterexample...

Thanks.

ss54, Feb 16, 2006
7. ### Victor S. MillerGuest

Gerald> A "prime triple" is the situation (p, p+2, p+6) where they are
Gerald> all primes. It is conjectured that there are infintely many
Gerald> prime triples.

Green and Tao have announced a proof of the prime k-tuple conjecture
-- see an article in the new Bulletin of the AMS. So this would
extend to any n by n matrix.

Victor

Victor S. Miller, Feb 22, 2006
8. ### Victor S. MillerGuest

Gerald> A "prime triple" is the situation (p, p+2, p+6) where they are
Gerald> all primes. It is conjectured that there are infintely many
Gerald> prime triples.

Victor> Green and Tao have announced a proof of the prime k-tuple
Victor> conjecture -- see an article in the new Bulletin of the AMS.
Victor> So this would extend to any n by n matrix.

I, of course, misspoke. While the Green and Tao result is a wonderful
result (that there are arbitrarily long arithmetic progressions
consisting only of primes), it is NOT the same as the prime k-tuple
conjecture! Mea Culpa!

Victor

Victor S. Miller, Feb 22, 2006