Is this all that surprising? The more symmetrically the primes are arranged around the interprime, the more accurate the approximation. There's nothing magical about the Gaussian: if you want more accuracy just replace (x-p_k)^2 with Abs[x-p_k]^q, q>2. Here's an easier way to locate the highly accurate interprimes: Table[ m = (Prime[n] + Prime[n + 1])/2; k = 1; While[n > k && (Prime[n - k] + Prime[n + 1 + k])/2 == m, k++]; m - Prime[n - k] , {n, 2, 100}] {2, 3, 6, 9, 10, 7, 4, 7, 19, 5, 8, 13, 4, 7, 9, 13, 5, 8, 5, 5, 10, 7, 10, 10, 13, 16, 7, 4, 13, 16, 7, 7, 17, 11, 5, 9, 34, 7, 9, 7, 13, 11, 14, 5, 8, 18, 14, 17, 4, 7, 7, 7, 13, 19, 9, 13, 5, 8, 5, 7, 17, 16, 29, 4, 11, 17, 11, 11, 4, 7, 14, 11, 9, 14, 7, 10, 16, 8, 13, 19, 17, 11, 5, 10, 7, 10, 10, 13, 4, 10, 16, 22, 16, 10, 7, 12, 13, 11, 21} This just gives the distance between the nth interprime and the furthest symmetrical prime. The largest element in this list, 34, corresponds to the fist occurrence of 444 digits of accuracy. Veit On Feb 15, 2009, at 5:42 AM, Ed Pegg Jr wrote:
Daniel Tisdale (daniel6874@gmail.com) sent MathWorld an interesting item.
"The sum $f(x)=sum_(k=1)^infty(-1)^ke^(-(x-p_k)^2)$ has as zeros precisely the interprimes, with the single additional point 5/2"
The plot agreed precisely, so I added it. http://mathworld.wolfram.com/Interprime.html
John Pardon (jpardon@princeton.edu) pointed out that it's wrong... that it gives a series of highly accurate approximations instead.
How accurate? f[x_] := N[Sum[(-1)^k*Exp[-(x-Prime[k])^2],{k,1,2500}],1000] Table[Floor[Log[10, Abs[1/f[Prime[n]/2 + Prime[n + 1]/2]]]], {n, 1, 400}]
2, 1, 3, 15, 35, 43, 21, 6, 10, 156, 10, 6, 73, 6, 21, 10, 52, 10, 6, 10, 10, 43, 21, 27, 6, 21, 111, 21, 6, 73, 27, 10, 21, 125, 21, 10, 21, 444 ...
The accuracies to the 400th Interprime have the following distribution: {{1, 1}, {2, 1}, {3, 1}, {6, 44}, {10, 61}, {15, 1}, {21, 70}, {27, 47}, {35, 23}, {43, 24}, {52, 32}, {62, 6}, {73, 23}, {85,13}, {111, 14}, {125, 19}, {156, 10}, {173, 1}, {210, 1}, {229, 2}, {271, 2}, {340, 1}, {417, 1}, {444, 2}}
Just the accuracies, in numbers of digits {1, 2, 3, 6, 10, 15, 21, 27, 35, 43, 52, 62, 73, 85, 111, 125, 156, 173, 210, 229, 271, 340, 417, 444}
Differences: 1, 1, 3, 4, 5, 6, 6, 8, 8, 9, 10, 11, 12, 26, 14, 31, 17, 37, 19, 42, 69, 77, 27
For starters, 444 digits of accuracy has occurred more than once. 52 digits of accuracy has occurred 32 times. Something seems to be happening here.
Ed Pegg Jr
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