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

 

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.

***

 

---