Problems & Puzzles: Puzzles

Puzzle 254.  Z=P2 - Q2

Someone self-named 'King' has been submitting to the well know site maintained by Caldwell and Honaker (Prime Curios!), several entries about the same  issue: numbers expressible, in K ways, as a difference of the square of consecutive primes.

As a matter of fact 'he' has obtained the earliest Z values for the first four K cases: 5, 72, 1848 and 4920.

Example: K=4, Z=4920:

4920 = 12312 - 12292 = 6172 - 6132 = 2512 - 2412 = 2112 - 1992

Using a smart approach you should be able to get larger entries, like the fifth member of the sequence, obtained by me in some few minutes using a simple code: 187117320.

K=5, Z= 187117320

187117320 = 467793312 - 467793292 =155931132 - 155931072 =77965612 - 77965492 = 46779432 - 46779232 = 31186372 - 31186072

Q. Find three more entries for this sequence.

Solution:

J. C. Rosa wrote (Feb. 2004)

I have only found three more solutions for K=5 : Z=277175640 ; Z=505270920;Z=837284520.

I have found a solution for K=6 but I am not sure that it is the earliest. Here it is :

Z=17036037480=2.2.2.3.5.7.11.337.5471
=2129504687^2-2129504683^2
=608429917^2-608429903^2
=425900947^2-425900927^2
=387182681^2-387182659^2
=304214969^2-304214941^2
=193591357^2-193591313^2

Later (April 2004) he added again:

I confirm that the number Z=17036037480 ( already published ) is the earliest Z value for K=6.
For Z=<33000000000 , I have not found any other solution for K=6. But I continue the search...

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