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| In [[number theory]], '''Brocard's conjecture''' is a [[conjecture]] that there are at least four [[prime number]]s between (''p''<sub>''n''</sub>)<sup>2</sup> and (''p''<sub>''n''+1</sub>)<sup>2</sup>, for ''n'' > 1, where ''p''<sub>''n''</sub> is the ''n''<sup>th</sup> prime number.<ref>{{mathworld|urlname=BrocardsConjecture|title=Brocard's Conjecture}}</ref> It is widely believed that this conjecture is true. However, it remains unproven as of January 2014.
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| {| class="wikitable" style="float:right; margin:10px;"
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| ! n !! <math>p_n</math> !! <math>p_n^2</math> !! Prime numbers !! <math>\Delta</math>
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| | 1 || 2 || 4 || 5, 7 || '''2'''
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| | 2 || 3 || 9 || 11, 13, 17, 19, 23 || 5
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| | 3 || 5 || 25 || 29, 31, 37, 41, 43, 47 || 6
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| | 4 || 7 || 49 || 53, 59, 61, 67, 71… || 15
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| | 5 || 11 || 121 || 127, 131, 137, 139, 149… || 9
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| |colspan=5| <math>\Delta</math> stands for <math>\pi(p_{n+1}^2) - \pi(p_n^2)</math>.
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| |}
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| The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... {{OEIS2C|id=A050216}}.
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| [[Legendre's conjecture]] that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for ''p''<sub>''n''</sub> ≥ 3 since ''p''<sub>''n''+1</sub> - ''p''<sub>''n''</sub> ≥ 2.
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| ==Notes==
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| {{reflist}}
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| {{Numtheory-stub}}
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| [[Category:Conjectures about prime numbers]]
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