Geodesic
Divisibility of Fermat Quotients
Matematičeskie zametki, Tome 92 (2012) no. 1, pp. 116-122.

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For any $\varepsilon >0$ and all primes $p$, with the exception of primes from a set with relative zero density, there exists a natural number $a\le(\log p)^{3/2+\varepsilon}$ for which the congruence $a^{p-1}\equiv 1 \,(\operatorname{mod} p^{2})$ does not hold.
Mots-clés : Fermat quotient
Keywords: smooth number, coset, oriented graph. 
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Yu. N. Shteinikov. Divisibility of Fermat Quotients. Matematičeskie zametki, Tome 92 (2012) no. 1, pp. 116-122. http://geodesic.mathdoc.fr/item/MZM_2012_92_1_a10/

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