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Maier's matrix method
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<h2 id="the-method">The method</h2> <p>The method first selects a <a href="/facts/Primorial/NqlQBLDp">primorial</a> and then constructs an interval in which the distribution of integers coprime to the primorial is well understood. By looking at copies of the interval translated by multiples of the primorial an array (or matrix) of integers is formed where the rows are the translated intervals and the columns are <a href="/facts/Arithmetic_progressions/c4ew9Xoq">arithmetic progressions</a> where the difference is the primorial. By <a href="/facts/Dirichlet%2527s_theorem_on_arithmetic_progressions/yFyJaE2d">Dirichlet's theorem on arithmetic progressions</a> the columns will contain many primes if and only if the integer in the original interval was coprime to the primorial. Good estimates for the number of small primes in these progressions due to (Gallagher 1970) allows the estimation of the primes in the matrix which guarantees the existence of at least one row or interval with at least a certain number of primes. </p> <ul><li><a href="/facts/Helmut_Maier/n4b29Yc8">Maier, Helmut</a> (1985), "Primes in short intervals", <i>The Michigan Mathematical Journal</i>, 32 (2): 221–225, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1307%2Fmmj%2F1029003189">10.1307/mmj/1029003189</a></li> <li>Maier, Helmut (1981), "Chains of large gaps between consecutive primes", <i><a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a></i>, 39 (3): 257–269, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0001-8708%2881%2990003-7">10.1016/0001-8708(81)90003-7</a></li> <li><a href="/facts/Patrick_X._Gallagher/oRlkgUXc">Gallagher, Patrick</a> (1970), "A large sieve density estimate near σ=1", <i>Inventiones Mathematicae</i>, 11 (4): 329–339, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1970InMat..11..329G">1970InMat..11..329G</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01403187">10.1007/BF01403187</a></li></ul>