Integer solution for $x^ay^b-z^ct^d=1$.

Question

P1 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a5^b-3^c11^d=1$.

P2 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a3^b-5^c7^d=1$.

I am looking for problems like these two problems.

Answer 1

I do not think a certain algorithm can be formulated. But what we can be sure of is that the product of powers of two primes(it seems you insist on x,y,z and t to be primes) plus or minus one is an even number that could be factorized to powers of some primes; for example:

$89\times3^2-1=800=5^2\times2^5$or $89^1 . 3^1-2^5 . 5^2=1$

$33^1 . 2^4 -31^1 . 17^1=1$

But if do not insist on primes then there can be infinitely many solutions because any even number can be composited to various powers of composites or a mixture of composite and prime factors; for example:

$2^8 . 31^2 - 3^2 . 27335^1=1$

Finding these sort of set of numbers is easy; take an arbitrary number(prime or no prime) multiply it by an even or odd number,subtract it by 1 and composite the result. Example:

$p=83; 83\times2 -1=65=5\times23⇒ 83\times2-65=5\times23=1; a=b=c=d=1$

$121\times49-1=8\times741⇒ 11^2.7^2-2^3.741^1=1; a=2,b=2,c=3,d=1$

$3^5.11^3-1=2^3.40429^1$

A clever program may help to find numbers which are all primes.