1,1

When k=j=2 P is a Mersenne prime 2^(n+1)-1 with n+1 prime As n increases sum(T(n,2)) for i=1 to n / sum(n^0.5) for i=1 to n tends to 0.62...... As n increases sum(T(n,1)) for i=1 to n / sum(n^0.5) for i=1 to n tends to 0.85...... As n increases sum(T(n,1)) for i=1 to n / sum(T(n,2)) for i=1 to n tends to 0.85....../0.62.....

Statistics on Pierre Cami's Riesel-hypotenuse (RH) primes I report a statistical study of primes, with the Riesel form k*j^n - 1, for j > 1, k > 0, 3001 > n > 0, occurring on hypotenuses of right-angled triangles in the (j,k) plane when the triangles yield no Riesel prime in their interiors.

For each integer n > 0, Pierre Cami defined the integers j(n) > 1 and k(n) > 0 by the following three requirements: P(n) = k(n)*j(n)^n - 1 is prime; k*j^n - 1 is not prime if j + k < s(n) = j(n) + k(n); k*j^n - 1 is not prime if j + k = s(n) and j < j(n). It is important to note that k(n) may contain a power of j(n).

Viewing this geometrically, we see that s(n) identifies a diagonal line in the (j,k) plane. Then the vertical line with the excluded Riesel number-base j = 1, the horizontal line with the excluded Riesel coefficient k = 0, and a hypotenuse with j + k = s(n), define a right-angled triangle within which there is no integer point (j,k) that yields k*j^n - 1 as a Riesel prime.

Finally, P(n) is the Riesel prime on the hypotenuse with a number-base j(n) smaller than that for any other prime on the hypotenuse. I shall refer to P(n) as the n-th Riesel-hypotenuse (RH) prime. The assonance with an alternative meaning of the acronym RH is deliberate and should cause no confusion in this report.

The sequence of Riesel-hypotenuse primes begins 3, 3, 7, 31, 31, 127, 127, 13121, 524287, 5119, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 6973568801 ... Cami remarked that every Mersenne prime occurs at least once as P(n) = M(n) = 2^n - 1. Only M(3) = 7 fails to appear previously as P(n-1) = 2*2^(n-1) - 1. Note that M(19) = 524287 also appears as P(9) = 2*4^9 - 1.

The non-Mersenne RH prime with smallest index is P(8) = 2*3^8 - 1 = 13121, since 3*2^8 - 1 = 13*59 is composite, as clearly are 2*2^8 - 1, 3^8 - 1 and 2^8 - 1. The smallest non-Mersenne RH prime is P(10) = 5*2^10 - 1 = 5119. At another extreme, we know that 3*2^4235414 - 1 is a 1274988-digit RH prime. Cami suggested comparing the sum of linear dimensions S(N) = sum(n=1,N,s(n)) with the sum of square roots S_1(N) = sum(n=1,N,sqrt(n)) conjecturing, from rather limited data, that the ratio S(N)/S_1(N) might tend to a constant as N tends to infinity. I remarked that his conjecture appeared to conflict with the prime number theorem (PNT) and suggested that S_2(N) = S_1(N)*sqrt(log(N)) might be a better candidate for comparison with S(N).

The geometrical rationale is that, on average, Cami's conjecture allowed merely O(N) attempts to find a prime P(N) = O(sqrt(N^(N+1))) by exploring a triangular area A = O(s(N)^2) of the (j,k) plane. The PNT suggests that we will need O(N*log(N)) attempts. Hence we should expect a linear size s(N) = O(sqrt(N*log(N))). I was unable to prove that S(N)/S_2(N) tends to a finite constant since I could not exclude the possibility of a weak polynomial dependence on sqrt(log(log(N))).

Using PFGW and GP script file, "pcs.txt" and "pcsp.gp", in http://physics.open.ac.uk/~dbroadhu/cert/pcrh.zip I determined the first 3000 RH primes, via the command line $ pfgw -f pcs.txt which produced GP-readable output of small proved primes in pfgw-prime.log and larger probable primes in pfgw.log. Combining these in "pcs.out", I invoked the GP script "pcsp.gp", which prints a table of [N, S/S_1, S/S_2] for N = 300 to 3000, in steps of 300, and writes 3000 candidate primes to "pcs.in", using ABC format for [k, j, n].

It computes the number of decimal digits, namely 7175, for the largest of these, 5*248^2996 - 1, and the number of times, namely 6750691, that the PRP command was invoked by the PFGW script.

The PFGW flag -f, to force trial division, ensured that the number of pseudoprimality tests was considerably smaller than this. Finally, BLS primality proofs were obtained via $ pfgw -lproofs.txt -tp pcs.in with certificates written to "proofs.txt", in the zipfile above. The results of this analysis, in "pcsp.out", are as follows: [ N, S/S1, S/S2] [ 300, 1.2714, 0.53235] [ 600, 1.3519, 0.53451] [ 900, 1.3804, 0.52925] [1200, 1.4243, 0.53492] [1500, 1.4530, 0.53729] [1800, 1.4609, 0.53360] [2100, 1.4695, 0.53131] [2400, 1.5035, 0.53892] [2700, 1.5034, 0.53487] [3000, 1.5155, 0.53558]

In conclusion, I remark that the second column of the table shows the slow growth expected from the prime number theorem and that the third column lends support to an approximation sum(n=1,N,j(n)+k(n)) ~ 0.53*sqrt(log(N))*sum(n=1,N,sqrt(N)) with small variations attributable to the limited sampling.

I am grateful to Pierre Cami for informing the PrimeForm group of his novel construction and for collegial on-list discussion. David Broadhurst, 4 February 2009 [From Pierre CAMI (pierre-cami(AT)orange.fr), Feb 06 2009]

David Broadhurst, The first 3000 rows

2*2^1-1=3 prime so T(1,1)=2 T(1,2)=2 2*2^2-1=7 prime so T(2,1)=2 T(2,2)=2 2*2^3-1=15 composite 3*2^3-1=23 prime so T(3,1)=3 T(3,2)=2

Sequence in context: A119469 A127439 A092788 this_sequence A091267 A003643 A058062

Adjacent sequences: A156048 A156049 A156050 this_sequence A156052 A156053 A156054

nonn,tabf

Pierre CAMI (pierre-cami(AT)orange.fr), Feb 02 2009