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- Durham University United Kingdom

as x ~ ~ , for which as for many other interesting results in the theory of numbers we are indebted to Chebyshev, has a t t racted the interest of several mathematicians. Revealed posthumously as little more than a fragment in one of Chebyshev's manuscripts, the theorem was first published and fully proved in a memoir by Markov in 1895 [6], while later in the same year a generalisation by Ivanov [4] appeared in which the polynomial n2§ 1 was replaced by n2+A for any positive A (an account of both Markov's and Ivanov ' s work is to be found in Paragraphs 147 and 149 of Landau's Primzahlen [5]). In 1921 Nagell [7] improved and further generalised Chebyshev's theorem by shewing that for any e log~ x, X