3, 7, 31, 127
 

What do these have in common?

This list may very well be complete.

[QUOTE=petrw1;512504]What do these have in common?

This list may very well be complete.[/QUOTE]Things that make me go MM

Only known instance where four consecutive primes, p, for which 2[SUP]p[/SUP]-1 is prime?

That is, where p[SUB]n[/SUB] is the nth prime and M(x) represents 2[SUP]x[/SUP]-1, then:

M(p[SUB]n[/SUB])
M(p[SUB]n+1[/SUB])
M(p[SUB]n+2[/SUB])
M(p[SUB]n+3[/SUB])

are all prime.

[QUOTE=petrw1;512504]What do these have in common?

This list may very well be complete.[/QUOTE]
Exponents of double Mersenne primes ( = Mersenne primes p such that 2^p - 1 is also a Mersenne prime).

Mersenne numbers that are also lucky numbers.

I thought this was too easy
 

The only 4 numbers so far that are:
- exponents of MPs
- an MP

7 for example:

[TEX]2^3-1 = 7[/TEX]
and
[TEX]2^7-1 = 31[/TEX]

The next candidate may be a few years away:
(TEX wasn't behaving here)

2[SUP]31[/SUP] = 2147483647
2[SUP]2147483647[/SUP] -1 = xxx

Is xxx a prime?

[QUOTE=petrw1;512568]The only 4 numbers so far that are:
- exponents of MPs
- an MP[/QUOTE]I think lukerichards also gave a plausible answer.

[QUOTE=retina;512570]I think lukerichards also gave a plausible answer.[/QUOTE]

For sure, my oversight....and Serge too.

[QUOTE=petrw1;512568]
The next candidate may be a few years away:
(TEX wasn't behaving here)

2[SUP]31[/SUP] = 2147483647
[/QUOTE]

I find TEX doesn't like superscripts of more than 1 digit unless you put them in curly braces. e.g.:

[CODE] [TEXX]2^{31}-1[/TEXX] [/CODE]

(I've slipped an extra x into the tag to stop it rendering TEX within the code block.)

Which gives:

[TEX]2^{31}-1[/TEX]