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In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r > g . This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive kdifferential on an hyperelliptic curve of genus g with a unique zero of order k(2g −2) for every (k, g ) 6= (2, 2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n > 2g . [ABSTRACT FROM AUTHOR]