It's fairly easy to see that the only sexy prime quintuple will be the one that contains 5 (see last part of the Wikipedia page) by imagining the number line as follows.<p>Now line up multiples of 5 and see if it's possible to fit them around these primes. There's always one multiple of 5 that lines up with a prime in the quintuple and hence one prime must be divisible by 5 and the only prime number divisible by 5 is 5.<p><pre><code> p.....p.....p.....p.....p
+ + + +
6 1 1 2
2 8 4
5....5....5....5....5....5
.5....5....5....5....5....
..5....5....5....5....5...
...5....5....5....5....5..
....5....5....5....5....5.
</code></pre>
Another way to think of that is as follows. Suppose that p is not divisible by 5 then one of p+1, p+2, p+3, p+4 must be. But p+6 = p+1+5, p+12=p+2+10, p+18=p+3+15, p+24=p+4+20, so whichever one it is there's a prime that's also divisible by 5. Similar reasoning works from the other numbers.
Of course, everyone knows the sexy primes are the 35/1.4, 50/1.4, 85/1.4 and 135/2.8<p>Sorry but photographic prime lenses beat mathematical prime numbers!