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<p>Create Project Euler/229 - Four Representations Using Squares (via create-page on MediaWiki MCP Server)</p> <p><b>New page</b></p><div>==Problem Statement==<br /> <br /> &#039;&#039;&#039;Four Representations Using Squares&#039;&#039;&#039;<br /> <br /> Consider the number 3600. It is very special, because<br /> <br /> 3600 = 48^2 + 36^2<br /> 3600 = 20^2 + 2×40^2<br /> 3600 = 30^2 + 3×30^2<br /> 3600 = 45^2 + 7×15^2<br /> <br /> Similarly, we find that 88201 = 99^2 + 280^2 = 287^2 + 2×54^2 = 283^2 + 3×52^2 = 197^2 + 7×84^2.<br /> <br /> In 1747, Euler proved which numbers are representable as a sum of two squares. We are interested in the numbers n which admit representations of all of the following four types:<br /> <br /> n = a_1^2 + b_1^2<br /> n = a_2^2 + 2 b_2^2<br /> n = a_3^2 + 3 b_3^2<br /> n = a_7^2 + 7 b_7^2,<br /> <br /> where the a_k and b_k are positive integers.<br /> <br /> There are 75373 such numbers that do not exceed 10^7.<br /> <br /> How many such numbers are there that do not exceed 2×10^9?<br /> <br /> ==Flags==<br /> <br /> {{ProjectEulerFlag}}</div>

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