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.

Later (March 2004) he added:

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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