Consider the Pell equation $x^2-p_n y^2=1$, where $p_n$ is the $n$th prime. Is $n^{2 \sqrt{n}}$ a reasonable upper bound for the smallest integer solution for $y$?
Above is a plot of $\log x$ (smallest solution for $y$) against $2 \sqrt{n} \log (n)$, plotted with
PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf =
ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[OddQ[n], n = 2*n];
s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]];
With[{n = 10^4}, Show[ListPlot[Log@(PellSolve[#] & /@
Prime@Range@n)[[All, 2]], PlotRange -> All], Plot[{2 Log@x Sqrt@x},
{x, 0, n}], PlotRange -> All]]
and
ListPlot[With[{n = 3*10^4}, ((Log@(PellSolve[#] & /@
Prime@Range[3, n])[[All, 2]])/(2 Log@# Sqrt@# & /@ Range[3, n]))],
PlotRange -> All]
Update
Using @WillJagy's data produces the following plot:
(N@With[{a=PrimePi@#},Log[PellSolve[#]][[2]]/(2Log@aSqrt@a)]&/@{5,13,29,53,61,109,181,277,397,409,421,541,661,1021,1069,1381,1549,1621,2389,3061,3469,4621,4789,4909,5581,6301,6829,8269,8941,9949,12541,13381,16069,17341,24049,24229,25309,29269,30781,32341,36061,39901,40429,43261,56149,58909,60589,63781,74869,82021,92821,107101,115021,125101,136189,140221,144589,145861,169789,199021,202981,216661,219829,226669,231709,245389,259429,266029,271021,300301,330229,379549,395509,448309,451669,455701,492781,548461,596821,602221,611389,617269,664381,679669,731509,759301,774229,799621,952429})//ListPlot
Added
Following up from @GregMartin's comment below, I add a plot coloured orange for primes congruent to $1 \!\!\mod 4$ and blue for primes congruent to $3 \!\!\mod 4$:
I ran some numbers using the prime itself and using the smaller number, $y,$ from $x^2 - p y^2 = 1.$ Oh, I only printed out $p$ when $y$ increased. Also $x \approx y \sqrt p.$
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Without printing $y$ itself, it gets very long:
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