1995 proposal BY Titu

Let $95n + 1 = { x }^{ 2 }$ and $19n + 1 = { y }^{ 2 }$ , for some positive integers $x$ and $y$ . Then ${ x }^{ 2 } - 5{ y }^{ 2 } = -4$ , which is generalized Pell equation with solutions given by

$\huge\ \frac { { x }_{ n } \pm { y }_{ n }\sqrt { 5 } }{ 2 } = { \left( \frac { 1\pm \sqrt { 5 } }{ 2 } \right) }^{ n }$ , $n = 1, 3, 5, ...$ .

Using the general formula for the Fibonacci sequence $\large\ { \left\{ { F }_{ k } \right\} }_{ k \ge 0 }$ , we conclude that $\large\ { y }_{ m } = { F }_{ 2m - 1 }$ , for $m = 1, 2, 3,...$ . It suffices to find the first term of the sequence $2, 5, 13, 34, 89, 233, 610, 1597, ...$ whose square is congruent to ${1} \pmod { 19 }$ . This term is ${ F }_{ 17 } = 1597$ , so the answer to the problem is

$\large\ n = \frac { 1 }{ 19 } \left( { { F }_{ 17 } }^{ 2 } - 1 \right) = \boxed{134232}$ .