Problems & Puzzles: Conjectures

Conjecture 64. NUMBER OF PRIME TWINS

Werner Sand sent the following conjecture:

Let p,q (p<q) be a prime twin.

Let “the sum of prime twins” be the sum of the upper partners q.

Let π2(x) be the number of prime twins below x.

Conjecture:

The number of prime twins below x is asymptotically equal to half the sum of prime twins below the square root of x:

π2(x) ~ ½ sum q (q=5..sqrt x) or:

The number of prime twins below x² is asymptotically equal to half the sum of prime twins below x:

π2(x²) ~ ½ sum q (q=5..x)

Generalization to triplets… k-tuples (constellations) see here.

Q. Can you prove it?

Enoch Haga wrote:

In A093683 I have number of twin primes under 10^n
In A139677 we have the approximate sums
See below: last number is sqrt of ~sums
4, 32, 5.66 Over
25, 820, 28.64 Over
174, 24676, 157.09 Under
1270, 1761248, 1327.12 Over
10250, 109650716, 10471.42 Over 86027, 7482340880, 86500.53 Over
738597, 543121286660, 736967.63 Under
6497407, 41216742789192, 6420026.70 Under (about 1.2%

The computation supports the conjecture.

***

Dana Jacobsen wrote on December 2014.

I found the comment by Enoch Haga to be confusing.  He's using A093683 (number of pairs of twin primes <= 10^n-th prime) for the first number, then matching it to sqrt(A139677) = sqrt(4*Pi2(10^{2*n})).  This doesn't seem to match the conjecture.  The conjecture involves summing twin primes below the square root of x, not taking the square root of a sum.  The range of the two sequences he's using is also different (10^n-th prime vs. 10^n).

I used:

perl -Mntheory=:all -E 'for my \$e (2 .. 18) { my \$n = int(10**\$e); my \$tpc = twin_prime_count(\$n); my \$tpca = twin_prime_count_approx(\$n); my \$sum = 0.5 * vecsum(map { \$_+2 } @{twin_primes(int(sqrt(\$n)))}); printf "n 10^%2d: %16d %14.9f  %14.9f\n", \$e, \$tpc, (\$tpc-\$tpca)/\$tpc, (\$tpc-\$sum)/\$tpc; }'

which, for n=10^e for e=2 to 18, looks at the actual twin prime count, the standard approximation (conjecture B of Hardy and Littlewood 1922), and Sand's conjecture.  We see:

n 10^ 2:                8    0.000000000     0.250000000
n 10^ 3:               35    0.000000000    -0.071428571
n 10^ 4:              205    0.009756098     0.385365854
n 10^ 5:             1224    0.001633987    -0.088235294
n 10^ 6:             8169   -0.001468968     0.256151304
n 10^ 7:            58980    0.003814852     0.106943032
n 10^ 8:           440312   -0.000129454     0.019546367
n 10^ 9:          3424506   -0.000234778     0.033362330
n 10^10:         27412679    0.000046001     0.004867273
n 10^11:        224376048    0.000032009     0.008664176
n 10^12:       1870585220    0.000013553     0.007745957
n 10^13:      15834664872    0.000004204    -0.004162089
n 10^14:     135780321665    0.000000418    -0.004708760
n 10^15:    1177209242304    0.000000637     0.000417552
n 10^16:   10304195697298    0.000000305     0.000267058
n 10^17:   90948839353159    0.000000067     0.000673782
n 10^18:  808675888577436   -0.000000016     0.000066098

showing (1) the percentage error is falling with Sand's conjecture, so it looks reasonable through 10^18, and (2) it is consistently worse than the Hardy/Littlewood approximation.  This is shown more clearly by looking at the values at 2^e.

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