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Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).

The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$). Jacobson, Lukes and Williams report on extensive calculations on regulators, and at the end of the paper state the belief that the fundamental unit can get as large as $\exp(c \sqrt{d}\log \log d)$; however, as also noted there, unconditionally we only know that the fundamental unit (or the solution in Pell's equation) sometimes gets as large as $\exp(c(\log d)^4)$ -- so there is a very large gap in our understanding. (See also my answer to the related MO question Upper bound for class number of a real quadratic field .)

Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).

The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$). Jacobson, Lukes and Williams report on extensive calculations on regulators, and at the end of the paper state the belief that the fundamental unit can get as large as $\exp(c \sqrt{d}\log \log d)$; however, as also noted there, unconditionally we only know that the fundamental unit (or the solution in Pell's equation) sometimes gets as large as $\exp(c(\log d)^4)$ -- so there is a very large gap in our understanding. (See also my answer to the related MO question Upper bound for class number of a real quadratic field .)

Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).

The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$).

Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).

The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$). Jacobson, Lukes and Williams report on extensive calculations on regulators, and at the end of the paper state the belief that the fundamental unit can get as large as $\exp(c \sqrt{d}\log \log d)$; however, as also noted there, unconditionally we only know that the fundamental unit (or the solution in Pell's equation) sometimes gets as large as $\exp(c(\log d)^4)$ -- so there is a very large gap in our understanding. (See also my answer to the related MO question Upper bound for class number of a real quadratic field .)

The accepted answer is not entirely accurate. There is a loss of precision in exponentiating the Brauer-Siegel limit which is not properly kept track of. Here is a fleshed out argument: As noted in that answer, the class number formula gives (with $\epsilon_p$ denoting the fundamental unit) $$ \log \epsilon_p = \sqrt{p} L(1,\chi)/h(p) \le \sqrt{p} L(1,\chi), $$ where $h(p) \ge 1$ is the class number. Now it is known that $L(1,\chi) \le C \log p$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $p$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). This would be enough to give your conjecture of $p^{C\sqrt{p}}$. On GRH we expect that $L(1,\chi)$ is bounded by $C\log \log p$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem, and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$).

Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).

The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$).