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Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?

Are there infinitely many? Finitely many? Something else?

(As usual, $\zeta_p$ denotes a primitive $p$-th root of unity.)

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It is expected that there are infinitely many primes for which it is $1$; indeed one expects this to hold for more than 70 per cent of all primes. However, this is open.

The exact value is only know for very few primes; I think only up to $67$ (this was true until some years ago, but perhaps I missed something).

There are however example where one know that it is somehow large, in particular larger than $p$. The first example for this is due to Cornell and Washington.

Recall that the Vandiver conjecture states it is not divisible by $p$, so that this size threshold is particularly interesting.

For more information (in particular details on the above heuristic) you could look at Schoof "Class numbers of real cyclotomic fields of prime conductor" Math Comp. (2003); this should be freely available.

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I thought it should be mentioned that van der Linden's result (which depends on Odlyzko's discriminant bounds) has recently been improved upon by a more careful accounting of the contribution of prime ideals to the explicit formula of the Dedekind zeta function of the Hilbert class field of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$.

The value of $h_p^+$ is now known up to 151 (and up to 241 under GRH). Please see http://arxiv.org/abs/1407.2373.

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