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I've never seen it ascribed to Shafarevich, but it is an old question. As a question, it equivalent to determining the period mod p of the sequence of Bell numbers discussed, e.g. in:

Levine, Jack; Dalton, R. E. Minimum periods, modulo p, of first-order Bell exponential integers. Math. Comp. 16 1962 416–423.

But they refer to even older papers. Any conjecture is wishful thinking since we can't get past $p=29$ or so with current technology. (Edit: As Kevin points out in the comments, I am seriously underestimating current technology, so this comment applies only to last century.)

Incidentally, I proved that the order is at least $2^{2.54p}$ in JTNB 16 (2004) 233-239.

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I've never seen it ascribed to Shafarevich, but it is an old question. As a question, it equivalent to determining the period mod p of the sequence of Bell numbers discussed, e.g. in:

Levine, Jack; Dalton, R. E. Minimum periods, modulo p, of first-order Bell exponential integers. Math. Comp. 16 1962 416–423.

But they refer to even older papers. Any conjecture is wishful thinking since we can't get past $p=29$ or so with current technology.

Incidentally, I proved that the order is at least $2^{2.54p}$ in JTNB 16 (2004) 233-239.