Geodesic
A Note on squares in arithmetic progressions, II
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 2, pp. 69-75 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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We show that the number of squares in an arithmetic progression of length $N$ is at most $c_{1}N^{3/5}(\log N)^{c_{2}}$, for certain absolute positive constants $c_{1}$, $c_{2}$. This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac{2}{3}$ in place of our $\frac{3}{5}$. The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1]. 
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     author = {Bombieri, Enrico and Zannier, Umberto},
     title = {A {Note} on squares in arithmetic progressions, {II}},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {69--75},
     year = {2002},
     volume = {Ser. 9, 13},
     number = {2},
     zbl = {1072.11010},
     mrnumber = {MR1949479},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_2_a0/}
}
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Bombieri, Enrico; Zannier, Umberto. A Note on squares in arithmetic progressions, II. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 2, pp. 69-75. http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_2_a0/