查看“三立方数和”的源代码

{{unsolved|数学|模9不同余4或5的整数是否都可以写成三整数立方之和？}}
[[File:Sum_of_3_cubes.svg|thumb|250px|整数''x''、''y''与''z''满足{{nowrap|1=''x''&#179; + ''y''&#179; + ''z''&#179; = ''n''}}的半对数图线，其中{{nowrap|''n'' ∈ [0, 100]}}。绿色条带代表已证明无解的整数。]]

'''三立方数和问题'''（{{lang-en|sums of three cubes}}）是指[[丢番图方程]]<math>x^3+y^3+z^3=n</math>是否存在整数解的问题。由于[[立方数]]模9同余0、1或-1，三立方数和模9不可能同余4或5，因而这是整数解存在的一个[[必要条件]]。然而，对于该条件是否同时为[[充分条件]]目前仍未有定论。

== 小整数例 ==

<math>n=0</math>时，若存在非平凡的三立方解，则[[费马大定理]]找到反例。此时三个立方数中必有两个同号，经移项，就会出现两正整数立方和等于另一正整数立方的情况。由于[[欧拉]]早已证明幂次为3的费马大定理{{r|euler}}，在<math>n=0</math>时的三立方和只有如下平凡解：
:<math>a^3 + (-a)^3 + 0^3 = 0.</math>
<math>n=1,2</math>时，存在如下解系，有无数解：
:<math>(9b^4)^3+(3b-9b^4)^3+(1-9b^3)^3=1</math>
以及，
:<math>(1+6c^3)^3+(1-6c^3)^3+(-6c^2)^3=2.</math>
上述表示经缩放可得，任意立方数或立方数的二倍都有三立方和{{r|w08|m36}}。除上述表示外，<math>n=1</math>也有其他三立方和解系{{r|ad}}，<math>n=2</math>有如下著名解{{r|ad|hlr}}：
:<math>1214928^3 + 3480205^3 + (-3528875)^3 = 2, </math>
:<math>37404275617^3 + (-25282289375)^3 + (-33071554596)^3 = 2,</math>
:<math>3737830626090^3 + 1490220318001^3 + (-3815176160999)^3 = 2.</math>
然而，已经证明只在1和2处存在能被[[二次多项式]]参数化的解析表示{{r|m42}}。即便在<math>n=3</math>处，也没有参数化解系。{{le|路易斯·J·莫德尔|Louis J. Mordell}}在1953年写道，除了其小整数解，“我对其一无所知”，即：
:<math>1^3+1^3+1^3=4^3+4^3+(-5)^3=3</math>
“我”也不知道为什么这三个数都满足模9同余{{r|m53}}。{{as of|2019|9}}，上述两式仍然是<math>n=3</math>仅有的已知解{{r|lue}}。

== 计算结果 ==
1955年起，莫德尔（{{Lang|en|Mordell}}）等许多学者都尝试过使用计算机寻找该问题的解。{{r|mw|gls|hlr|cv|b|e|bpty|ej|h}}对于1000以内的正整数<math>n</math>，埃尔森汉斯（{{Lang|en|Elsenhans}}）与雅内尔（{{Lang|en|Jahnel}}）于2009年使用{{le|诺姆·埃尔奇斯|Noam Elkies}}提出的基于[[格规约]]的方法{{r|e}}找到了<math>\max(|x|,|y|,|z|)<10^{14}</math>范围内的所有解。2016年，于斯曼（{{Lang|en|Huisman}}）使用同样的方法将搜索上界提升至<math>\max(|x|,|y|,|z|)<10^{15}</math>。到此时为止，<math>n<100</math>的正整数中，33与42以外所有模9不同余4或5的<math>n</math>都找到了至少一组整数解。{{r|h}}

2019年，[[安德鲁·布克]]（{{Lang|en|Andrew Booker}}）采用一种新方法发现了<math>n=33</math>的一组解：{{r|arb}}

:<math>33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3</math>

此时，他在<math>\min(|x|,|y|,|z|)<10^{16}</math>的范围里尚没有找到<math>n=42</math>的解。{{r|arb}}

随后在2019年9月，布克和[[安德鲁·萨瑟兰]]（{{Lang|en|Andrew Sutherland}}）最终敲定了42的一个解，并在MIT数学系的网站上贴了出来{{NoteTag|流行文化中，42被称[[生命、宇宙以及任何事情的终极答案]]，萨瑟兰在页面的标题提到了这个典故：[http://math.mit.edu/~drew/ Life, The Universe, and Everything]}}：

:<math>42=(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3</math>

这个解的获得在Charity Engine全球网络（{{Lang|en|Charity Engine's global grid}}）上耗费了130万机时。

至此所有100以内的数都找到了解。{{as of|2019|9}}，未能求解最小整数是<math>n=114</math>{{r|lue}}，如果有解的話，<math>(|x|,|y|,|z|)</math>至少有一數大於[[100000000000]]。

仅剩的未解決的在[[1000]]以內的整数是[[114]]、[[390]]、[[579]]、627、633、732、921和975，一共有8個。

== 注释 ==
{{NoteFoot}}

== 参考文献 ==
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}}

[[Category:丢番图方程]]
[[Category:数学中未解决的问题]]