# Re: JSH: Math journals do not just die

Tm Peters wrote:

[josephus]

It is not intuitively obvious that the difference bwteen two primes is
an even number

Your intuition could use a tuneup ;-) These are the primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

Note that, except for 2, they're all odd: no even integer greater than
2 can be a prime, because every even number is divisible 2. And the
difference between two odd integers is always even (that has nothing
directly to do with primes, it applies to the difference of any odd
integers). Therefore if p and q are any two primes greater than 2, p-q
is even.

mea culpa. I am NOT A MATHEMATICIAN. but I figure if I tried I could not do worse than James Harris.
josephus.

if the sequence is mod 2

Sorry, don't know what that means. James's sequence is mod(p, 3) for
primes p > 3. No prime greater than 3 is divisible by 3 (else it
wouldn't be prime!), so 0 never appears in his sequence, only 1 and 2
appear.

the result of p mod 3 will be 2 and it it is 4 p mod 3 will be 1. so we could
have 0 , 1, or 2. some prime difference is a multilple of two?

As above, the difference between any two primes greater than 2 is a
multiple of 2, because all such differences are even.

marcus_b was talking about prime gaps because these things are true of
every two consecutive elements:

mod(p, 3), mod(q, 3)

of James's sequence:

The two elements are 1 and 2 (in that order) if and only if q-p leaves
a remainder of 4 when divided by 6.

The two elements are 2 and 1 (in that order) if and only if q-p leaves
a remainder of 2 when divided by 6.

The two elements are 1 and 1, or 2 and 2, if and only if q-p is evenly
divisible by 6.

There are clear reasons for why gaps divisible by 6 are less common
than the sum of the number of gaps leaving a remainder of 2 or 4 when
divided by 6 across primes with even dozens of digits, and that in turn
explains why the pairs 11 and 22 occur less often than the pairs 12 and
21. For example, to get a gap of exactly 6, p and p+6 both have to be
prime, but neither p+2 nor p+4 can be prime (otherwise the gap would be
2 or 4 instead of 6). At least across "small" primes, that makes a gap
of 6 less frequent than James needs it to be to hold a rational belief
that the sequence behaves anything even vaguely like a random sequence
behaves.

The degree to which the distribution of pairs "acts randomly" has been
rigorously measured across all primes less than a billion (10^9). The
probability that a truly random sequence could have produced a pair
distribution so unbalanced is so small it's probably beyond human
comprehension. For example, it's enormously more likely that you'll
win the lottery tomorrow /and/ be killed by lightning the day after --
and "enormously" is far too weak a word there.

If the entire observable universe were packed as solidly as possible
with electrons, and I picked one at random, then you picked one at
random, it's enormously more likely that you'd pick the same one.
That's based on:

http://en.wikipedia.org/wiki/Electron

Based on the classical electron radius and assuming a dense
sphere packing, it can be calculated that the number of electrons
that would fit in the observable universe is on the order of
10^130.

One chance in 10^130 is enormously, inconceivably larger than the one
chance in 10^60000 the simple pair test left James's sequence of being
truly random -- and that's actually cutting him a huge break, as the
calculated odds were actually close to 1 in 10^66510, and one chance in
10^60000 is enormously, inconceivably more likely than that.

OTOH, the observed distribution of pairs does closely follow what
80-year-old conjectures about the distribution of prime gaps predict.

.