Mark Underwood wrote:

> The 9 primes after 1260 are

>

> 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).

>

> I've checked up to six million and thus far the 9 string

> above is the longest cluster of, for lack of a better

> term, doubly consecutive primes.

I have found 3 cases of 19 "doubly consecutive primes".

I think they are the first known above 18.

n=66916817613475787 and n=304555583497054847 both give consecutive primes:

n + (0,2,6,12,20,62,66,126,132,140,144,146,150,152,180,192,210,242,276)

n=149888370336254401 and n=224561886371432461 both give:

n + (0,12,18,36,76,112,126,148,186,196,208,222,246,250,252,256,258,298,316)

n=56029782174606241 and n=267322720146562081 both give:

n + (0,28,58,66,70,96,100,102,106,108,130,148,156,162,240,276,312,336,378)

The same 3 in Mark's notation:

237638765883579060 + (66916817613475787, ..., 66916817613476063)

74673516035178060 + (149888370336254401, ..., 149888370336254717)

211292937971955840 + (56029782174606241, ..., 56029782174606619)

q=34837285 prp quintuplets were computed on the form:

k*17# + (97, 101, 103, 107, 109), for k<734445220000

This ensured q*(q-1)/2 ~= 6*10^14 pairs with at least

5 consecutive primes in the same pattern.

The use of 17# gave the same divisibility properties

for primes<=17 to increase odds of more matches.

The previous 14 and next 14 prp's were computed for each quintuplet.

Comparison gave 3 cases with 14 combined consecutive matches.

The GMP library made prp tests. PARI/GP proved the solution primes.

The DOS/Windows command "sort" was used on a 1.6 GB file during comparison.

(Has "sort" ever been credited in a prime record?)

--

Jens Kruse Andersen