Problems & Puzzles: Puzzles

Puzzle 967. Extending Puzzle 485.

Time ago, while solving the puzzle 485, Mr. Jacques Tramu found the following arithmetical expression, using the first 13 primes:

17*23*31*41 - 2*3*7*11*29*37 = 5*13*19

Q1. Can you find similar expressions using the first n>13 primes?

Q2. Can you find similar expressions using n>13 consecutive primes?


Contributions came from Adam Stinchcombe


Adam wrote on Aug 26, 2019:



Q2 is impossible. If you have a product of “large” primes minus a product of “large” primes then this equation will be of the form   (odd – odd) = even.  So you have to start with the first primes, starting with 2.



But using an updated threesome of summands I found algebraic identities:


p(p+12)+(p+18)+(p+30) = (p+6)(p+8) then I found p=4402241 yields consecutive primes


and    p(p+12)+(p+6)+(p+34)=(p+4)(p+10) with p=3457 and 101107 and many more values.


I found the identities with a little bit of logic (primes are of the form 6n+1,6n-1) and searching by hand through possibilities.  Then I ran the computer to find the specific gap sequence I wanted.


One might argue that a single factor does not a product make.



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