Problems & Puzzles: Puzzles

Puzzle 912. Two successive primes and oblong numbers.

In one Curio posted in his pages, G. L, Honaker, Jr. wrote that:

12 is The smallest oblong number that is the sum of 2 successive primes

An "oblong number" N is such that N=n*(n+1).

In this case 12 = 3*4 = 5+7

Here we ask for the successive primes p & q such that n*(n+1)=p+q, for a given N.

By my side I (CR) made some search just to see how difficult is to get these oblong numbers and the associated primes. It doesn't seems to be too hard to get large solutions. Here are a couple of examples:

Example 1:
n= 14142135623730950487637884
N=n*(n+1)= 199999999999999999989280183638997148141256399635340
P= 99999999999999999994640091819498574070628199817603 (50 digits)
Q= P+134

Example 2:
n= 1414213562373095048763788073031832936976570636034011840820312500000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000001199

N=n*(n*1)=
1999999999999999999892800770162982594503765590264720633949886640289254553910048
63993364981666900348500348627567291259765625000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000033912841225706819269355637991303353828698163852095603942871093
75000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000001437601

P= 99999999999999999994640038508149129725188279513236031697494332014462727695502
43199668249083345017425017431378364562988281250000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000001696349168066527510992163793601683607903396477922797203063964
84375000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000715933 (1000 digits)

Q-P=6934

Q. Send the successive primes for the largest oblong number you can compute.
 



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