Problems & Puzzles: Puzzles

Puzzle 460. Sqrt[p(x)*p(x+y)-1]= y

Sebastián Martín Ruiz sent the following puzzle:

Looking for solutions of this equation:

Sqrt[p(x)*p(x+y)-1]= y
p(x)=x-th prime
Sqrt=Square Root

I have found three (3) solutions:

x=1 y=5
x=3 y=22
x=10  y=186

Q. Find all solutions.


Luke Pebody wrote:

 have been running the following programme in Magma:
* Set p = 2, q = 3, y = 1 and N to be the empty set.
* Start loop
* Increase y, set q = NextPrime(q)
* Set z = y*y+1
* WHILE p>2 AND z < p*q, set p = PreviousPrime(p) and q = PreviousPrime(q)
* WHILE z > p*q, set p = NextPrime(p) and q = NextPrime(q)
* If p*q = z, add the element <p,q,y> to the set N
* Print <y,N>

The latest output is
<31260000, {<2,13,5>, <5,97,22>, <29,1193,186>}

Thus there is no solution with y<31 260 000


Enoch Haga wrote:

 I have verified that the three solutions in the problem are the only
ones through n=10^7.

The question suggests that solutions may be finite. If the sequence is infinite, then each new solution I think will be found at increasingly long distances.



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