\u95ee\u9898<\/p>\n
\\(a,b\u4e3a\u6b63\u6574\u6570\uff0c\u82e5\\frac{a^2+b^2}{ab+1}\u4e3a\u6574\u6570\uff0c\u6c42\u8bc1\u5176\u4e5f\u4e3a\u5e73\u65b9\u6570\u3002\\)<\/blockquote>\n\u89e3\u7b54<\/p>\n
\\(\u4e0d\u59a8\u8bbea\\geq b>0\uff0c\u4ee4k=\\frac{a^2+b^2}{ab+1}\uff0c\u5219\u6709a^2-bka+b^2-k=0\\)
\n\\(\u6784\u9020\u65b9\u7a0b\\)
\n\\(x^2-kbx+b^2-k=0\\)
\n\\(\u663e\u7136a\u4e3a\u65b9\u7a0b\u7684\u89e3\u3002\u8003\u8651\u65b9\u7a0b\u7684\u53e6\u4e00\u4e2a\u89e3\u3002\\)
\n(1) \u82e5\\(a=b\\)\uff1a\u5bb9\u6613\u6c42\u5f97\\(a=b=1, k=1\\)\u3002
\n(2) \u82e5\\(a>b\\)\uff1a
\n\\(a+x_2=kb\u4e3a\u6574\u6570\uff0c\u5219x_2\u4e3a\u6574\u6570\u3002\u4ee4x_2=a’\\)
\n\\(\u5148\u8bc1a’\u975e\u8d1f\u3002\u82e5a’\\leq-1\uff0c\\)
\n\\(\u5219{a’}^2+b^2=k(a’b+1)\\leq k(1-b)\\leq0\uff0c\u4e0eb\u662f\u6b63\u6574\u6570\u77db\u76fe\u3002\\)
\n\\(\u56e0\u800ca’\u975e\u8d1f\u3002\\)
\n\\(\u7531\u4e8eaa’=b^2-k\uff0c\u5219\\)
\n\\(a’=\\frac{b^2-k}{a}<\\frac{b^2}{a}<\\frac{ab}{a}=b\\\\
\n\u4ee4a_1=b,b_1=a’\uff0c\u5219\u6709\\\\
\nb_1^2-ka_1b_1+a_1^2-k=0\\\\
\n\u5373k=\\frac{a^2+b^2}{ab+1}=\\frac{a_1^2+b_1^2}{a_1b_1+1}=\\cdots\\\\
\n\u800ca>b>b_1>b_2>\\cdots>b_n\\geq 0\\\\
\n\u7531\u4e8eb_n\u6709\u4e0b\u754c\u4e14\u975e\u8d1f\uff0c\u5fc5\u6709\u6700\u540e\u4e00\u4e2ab_n=0\\\\
\n\u6b64\u65f6k=a^2\u4e3a\u5e73\u65b9\u6570\u3002\\)
\n\u8bc1\u6bd5\u3002<\/p><\/blockquote>\n\u7f16\u7a0b\u53ef\u6c421000\u4ee5\u4e0b\u7684\\(a,b\\)<\/p>\n
In [5]: for a in range(1, 1001):
\n…: for b in range(a, 1001):
\n…: k = (a*a + b*b)\/(a*b + 1)
\n…: if k == int(k):
\n…: print(f'{a=}, {b=}, {k=}’)
\n…:
\na=1, b=1, k=1.0
\na=2, b=8, k=4.0
\na=3, b=27, k=9.0
\na=4, b=64, k=16.0
\na=5, b=125, k=25.0
\na=6, b=216, k=36.0
\na=7, b=343, k=49.0
\na=8, b=30, k=4.0
\na=8, b=512, k=64.0
\na=9, b=729, k=81.0
\na=10, b=1000, k=100.0
\na=27, b=240, k=9.0
\na=30, b=112, k=4.0
\na=112, b=418, k=4.0<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"\u95ee\u9898<\/p>\n
[latex]a,b\u4e3a\u6b63\u6574\u6570\uff0c\u82e5\\frac{a^2+b^2}{ab+1}\u4e3a\u6574\u6570\uff0c\u6c42\u8bc1\u5176\u4e5f\u4e3a\u5e73\u65b9\u6570\u3002[\/latex]<\/blockquote>\n\u89e3\u7b54<\/p>\n
[latex]\u4e0d\u59a8\u8bbea\\geq b>0\uff0c\u4ee4k=\\frac{a^2+b^2}{ab+1}\uff0c\u5219\u6709a^2-bka+b^2-k=0[\/latex]
\n[latex]\u6784\u9020\u65b9\u7a0b[\/latex]
\n[latex]x^2-kbx+b^2-k=0[\/latex]
\n[latex]\u663e\u7136a\u4e3a\u65b9\u7a0b\u7684\u89e3\u3002\u8003\u8651\u65b9\u7a0b\u7684\u53e6\u4e00\u4e2a\u89e3\u3002[\/latex]
\n(1) \u82e5[latex]a=b[\/latex]\uff1a\u5bb9\u6613\u6c42\u5f97[latex]a=b=1, k=1[\/latex]\u3002
\n(2) \u82e5[latex]a>b[\/latex]\uff1a
\n[latex]a+x_2=kb\u4e3a\u6574\u6570\uff0c\u5219x_2\u4e3a\u6574\u6570\u3002\u4ee4x_2=a’[\/latex]
\n[latex]\u5148\u8bc1a’\u975e\u8d1f\u3002\u82e5a’\\leq-1\uff0c[\/latex]
\n[latex]\u5219{a’}^2+b^2=k(a’b+1)\\leq k(1-b)\\leq0\uff0c\u4e0eb\u662f\u6b63\u6574\u6570\u77db\u76fe\u3002[\/latex]
\n[latex]\u56e0\u800ca’\u975e\u8d1f\u3002[\/latex]
\n[latex]\u7531\u4e8eaa’=b^2-k\uff0c\u5219[\/latex]
\n[latex]a’=\\frac{b^2-k}{a}<\\frac{b^2}{a}<\\frac{ab}{a}=b\\\\
\n\u4ee4a_1=b,b_1=a’\uff0c\u5219\u6709\\\\
\nb_1^2-ka_1b_1+a_1^2-k=0\\\\
\n\u5373k=\\frac{a^2+b^2}{ab+1}=\\frac{a_1^2+b_1^2}{a_1b_1+1}=\\cdots\\\\
\n\u800ca>b>b_1>b_2>\\cdots>b_n\\geq 0\\\\
\n\u7531\u4e8eb_n\u6709\u4e0b\u754c\u4e14\u975e\u8d1f\uff0c\u5fc5\u6709\u6700\u540e\u4e00\u4e2ab_n=0\\\\
\n\u6b64\u65f6k=a^2\u4e3a\u5e73\u65b9\u6570\u3002[\/latex]
\n\u8bc1\u6bd5\u3002<\/p><\/blockquote>\n\u7f16\u7a0b\u53ef\u6c421000\u4ee5\u4e0b\u7684[latex]a,b[\/latex]<\/p>\n
In [5]: for a in range(1, 1001):
\n…: for b in range(a, 1001):
\n…: k = (a*a + b*b)\/(a*b + 1)
\n…: if k == int(k):
\n…: print(f'{a=}, {b=}, {k=}’)
\n…:
\na=1, b=1, k=1.0
\na=2, b=8, k=4.0
\na=3, b=27, k=9.0
\na=4, b=64, k=16.0
\na=5, b=125, k=25.0
\na=6, b=216, k=36.0
\na=7, b=343, k=49.0
\na=8, b=30, k=4.0
\na=8, b=512, k=64.0
\na=9, b=729, k=81.0
\na=10, b=1000, k=100.0
\na=27, b=240, k=9.0
\na=30, b=112, k=4.0
\na=112, b=418, k=4.0<\/p><\/blockquote>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/posts\/442"}],"collection":[{"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=442"}],"version-history":[{"count":56,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/posts\/442\/revisions"}],"predecessor-version":[{"id":556,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=\/wp\/v2\/posts\/442\/revisions\/556"}],"wp:attachment":[{"href":"https:\/\/blog.messi.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=442"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=442"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.messi.moe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}