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.

Paul Cleary wrote on July 13, 2019

Here is the next solution after your example.

n=

14142135623730950487637880730318329369765706360340118408203125000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000007667.

N=n(n+1)=

19999999999999999998928007701629825945037655902647206339498866402892545539100
48639933649816669003485003486275672912597656250000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000002940008574817427296875039025025877492680592
69525110721588134765625000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000108045630.

p= 99999999999999999994640038508149129725188279513236031697494332014462727695
50243199668249083345017425017431378364562988281250000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000010843482489495706286396345049971579044267
85535179078578948974609375000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000293947
37.

Q-P=1082

This is the 17th solution after that one

n=

1414213562373095048763788073031832936976570636034011840820312500000000000000

0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000033521.

N=n(n+1)=

199999999999999999989280077016298259450376559026472...0000001123690962.

p=

999999999999999999946400385081491297251882795132360...0000000561845063.

Q-P=836.

Here is my largest so far

n=

141421356237309504876378807303183293697657063603401184082031250000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000002929.

N=n(n+1)=

19999999999999999998928007701629825945037655902647...0000000008581970.

p=

99999999999999999994640038508149129725188279513236...0000000004289737.  (1002 digits).

Q-p=2496

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