Let $a,b$ be positive integers. How to efficiently solve the diophantine equation
$$94 \cdot 10^{16} + 1 = a^2 + b^2$$
?
The solutions $a,b$ are unique (up to switching $a$ and $b$) because $94 \cdot 10^{16} + 1$ is a prime $1 \mod 4$.
Ofcourse one of $a,b$ is odd while the other is even. In fact $a,b$ are coprime.
A big search with a computer is ofcourse not considered valid.
Considering the expansion of
$$(1 + x + x^4 + x^9 + x^{16} + x^{25} + x^{36} + ...)^2$$
at the term $x^{94 \cdot 10^{16} + 1}$ without any speed up tricks is also not considered efficient.
With some guessing and puzzling and assuming that the ratio $a/b$ was not to boldly far from $94$ (*) I managed to get :
$$a = 969 427 940,b = 14 473 049 $$
where the ratio $a/b = 66.98..$
- $94$ because $10^{16} + 1 $ is close to a square and so is $94$ ????
Is it true that Ramanujan solved this type of diophantines out of his head with continued fractions ?
How to reduce this to a problem of continued fractions ?
Is there a useful closed form for $a,b$ ?
What number theoretical functions help alot here ? Quadratic residue ? Integer Factoring ? (if that is still considered efficiently is another matter)
Perhaps a line of attack is set $p = 94 \cdot 10^{16} + 1$ and for some integers $Q,S$ we have $Q p = S^2 + 1$ [*]
Then $S^2 + 1 = (S + i)(S - i)$ and {$GCD(S+i,p)$, $GCD(S-i,p)$} is equal to {$a + bi$,$a-bi$}.
CGD is easier than factoring for any euclidean ring, so both integers and gaussian integers.
But I have no efficient method to solve $Q p = S^2 + 1$.
Maybe trying pythagoras ideas like $Q p - 1 = S^2 = u^2 + v^2$, if that even makes sense ?
Should I involve cubics maybe ?
Or maybe some kind of "Chakravala method" ?
see :
https://en.wikipedia.org/wiki/Chakravala_method
Or I would be impressed by some kind of infinite descent, however that is usually used to show something does not exist or is unique in a nonconstructive way.
I seem unable to find a deterministic efficient algoritm.
