A one-sentence proof that every prime p Ξ1(mod 4) is a sum of two squares
A one-sentence proof that every prime p Ξ1(mod 4) is a sum of two squares
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一句话证明每个素数 p Ξ1(mod 4) 是两个平方和
DOI：
10.1080/00029890.1990.11995566
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发表时间：
1990
影响因子：
0.5
通讯作者：
D. Zagier
中科院分区：
文献类型：
--
作者：
D. Zagier
这个证明是由于Heath-Brown[1]的一个证明的简化(反过来，灵感来自Liouville给出的一个证明)。对隐含的断言的验证--S是有限的，地图是明确定义的和对合的(即，等于它自己的逆)并且恰好有一个固定点--是直接的，留给了读者。只有最后一种方法要求p是4k+1形式的素数，不动点为(1，1，k)。请注意，证明不是构造性的：它没有给出一种方法来实际找到p作为两个平方和的表示。用不动点定理证明的拓扑学和分析中的结果也出现了类似的现象。实际上，我们所使用的基本原理：“有限集和它的不动点集在任意对合下的基数具有相同的奇偶性”，是相应的拓扑结果的组合类比和特例：“一个拓扑空间和它的不动点集在任何连续对合下的欧拉特征具有相同的奇偶性。”
This proof is a simplification of one due to Heath-Brown [1](inspired, in turn, by a proof given by Liouville). The verifications of the implicitly made assertions-that S is finite and that the map is well-defined and involutory (ie, equal to its own inverse) and has exactly one fixed point-are immediate and have been left to the reader. Only the last requires that p be a prime of the form 4k+ 1, the fixed point then being (1, 1, k).Note that the proof is not constructive: it does not give a method to actually find the representation of p as a sum of two squares. A similar phenomenon occurs with results in topology and analysis that are proved using fixed-point theorems. Indeed, the basic principle we used:" The cardinalities of a finite set and of its fixed-point set under any involution have the same parity," is a combinatorial analogue and special case of the corresponding topological result:" The Euler characteristics of a topological space and of its fixed-point set under any continuous involution have the same parity."