Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first few primes of form $4k+1$ (equivalently, $\mathbb{Q}(\sqrt{p})$ having odd conductor)?
To clarify the question:


*

*A conceptual explanation should treat the first few primes simutaneously, instead of a case-by-case analysis where a case is a single prime. (A case-by-case analysis with a finite number of cases that a priori covers the whole range of primes is allowed, e.g. the cases being p=1, 5 or 9 mod 12.)

*For "the first few primes of form $4k+1$", I mean such continuous primes up to a bound, e.g. $5,13,17$ but not $5,17,29$. The argument should be able to cover primes in such a way.

*To avoid trivialities, the conceptual explanations should cover at least $5, 13, 17$ and $29$.
An example of conceptual explanation would be like:


*

*By Example 2.9 of Masley's paper Class numbers of real cyclic number fields with small conductor, the class number of such fields are odd.

*The Minkowski bound gives $h(\mathbb{Q}(\sqrt{p}))<3$ for $p<36$. Thus we have established $h(\mathbb{Q}(\sqrt{p}))=1$ for the first few primes of form $4k+1$: $5,13,17$ and $29$.

*This explanation also works for cyclic cubic fields of conductor $7$ and $13$. 
Bonus for explanations that are not specialized on real quadratic fields, e.g. the explanation presented above.
 A: Some complementary heuristics, too long for a comment.
Again start from the fact that the class number $h$ of $K = {\bf Q}(\sqrt{p})$
is odd if $p$ is a prime of the form $4k+1$.  This time we compare with Dirichlet's class number formula, which here gives
$$
L(1,\chi_p) = \frac{2\log \epsilon}{\sqrt p} h,
$$
where the character $\chi_p$ is the Legendre symbol $\chi_p(n) = (n/p)$,
and $\epsilon$ is the fundamental unit of $K$.
We expect that $L(1,\chi_p) \approx 1$, so large $h$ go with small $\epsilon$.
A unit $\epsilon > 1$ in a real quadratic field of discriminant $D$
must be at least as large as $\frac12(m + \sqrt{m^2 \pm 4})$
for some odd integer $m$, with $D = m^2 \pm 4$.
If $D$ is prime then we must use the plus sign
(unless $m=3$, but then the fundamental unit is $(1+\sqrt5)/2$).
$\epsilon > \sqrt{p} - O(1/\sqrt{p})$ and
So, $2\log \epsilon / \sqrt{p} > \log p - O(1/p)$.
Setting $L(1,\chi_p) \approx 1$ and $h=3$ in the class number formula gives
$\sqrt{p} \approx 3 \log p$; the solution $p \approx 289$ is of about
the right size for the minimal example of $h>1$.
We're actually closer here than we deserved to be:
the solution of $L(1,\chi_p) \sqrt{p} = 3 \log p$ is quite sensitive to
the size of $L(1,\chi_p)$, and $L(1,\chi_{229}^{\phantom.}) = 1.075+$
is unusually close to $1$; for example, $L(1,\chi_p) > 2$ for
$p = 193, 241, 313, 337$, while $L(1,\chi_p) < 0.4$ for
$p = 173, 293, 677, 773$.
Most of the early examples of $h > 1$ have small $\epsilon$,
either with $p=m^2+4$ as above or the next-smallest possibility,
$\epsilon = m + \sqrt{p}$ with $p=m^2+1$.  Indeed
this
LMFDB list of fields ${\bf Q}(\sqrt p)$ with $p<2000$ and $h>1$
begins with
$$
229 = 15^2 + 4,\
257 = 16^2 + 1,\
401 = 20^2 + 1,\
577 = 24^2 + 1,
$$
$733 = 27^2 + 4$,
and then two exceptions $p=761$ and $p=1009$ and nine further $p$
of which all but $1429, 1489, 1901$ are not of the form $m^2+4$ or $m^2+1$.
Moreover $229$ is the only second-smallest prime of the form $p = m^2 + 4$
that satisfies our analytic bound $p > 63$ --- and the smallest is
$p = 173$, which was our example of an unusually small $L(1,\chi_p)$.
Likewise the next two examples were $293 = 17^2 + 4$ and $677 = 26^2 + 1$,
which are conjectured to be the largest discriminants $p = m^2+4$ and $p = m^2+1$
for which ${\bf Q}(\sqrt p)$ has class number $1$.
A: We give a uniform approach to $p \leq 61$ by applying
analytic discriminant bounds to the Hilbert class field.
To be sure this is not entirely "conceptual", but then
some computation is needed even to deal with $p < 36$ using Minkowski.
If $p = 4k+1$ is prime then $K = {\bf Q}(\sqrt{p})$ has odd class number $h$,
so either $h=1$ or $h \geq 3$.  If $h \geq 3$ then the Hilbert class field
$H_K$ is a totally real field of degree $2h \geq 6$ and discriminant $p^h$.
We can now apply to $H_K$ the known lower bounds on the discriminants of
totally number fields.
The Odlyzko bounds (see Table 4 on page 134 of
his
1990 paper give a lower bound of $7.941$ on the root-discriminant
$|{\rm disc}(F)|^{1/n}$ for any totally real field $F$ of degree $n \geq 6$.
Hence $p > 7.941^2 > 63$.  So we have accounted for all $p \leq 61$.
If the zeta function $\zeta_F$ satisfies the Riemann hypothesis,
the lower bound improves to $8.143$.  Unfortunately this does not
account for any more primes because $8.143^2 = 66.3+$ and
$65$ is not prime.  Since there exists a totally real sextic field
of discriminant $300125 = 5^3 7^4$ (see the
LMFDB entry),
such bounds can never get us past $300125^{1/3} = 66.95+$,
so the primes $p \in [73, 197]$ must be dealt with in some other way.
