不定方程x^a+y^b=z^c的研究

朱明君

(第1题)
\(不定方程x^n+y^{\left( n+1\right)}=z^{\left( n+2\right)}的通解公式\)

\(一，设n为奇数，\)
\(\frac{n\left( n+2\right)+1}{2}=m，\)
\(则\left( 2^m\right)^n+\left( 2^{m-\left( \frac{n+1}{2}\right)}\right)^{n+1}=\left( 2^{m-n}\right)^{n+2}\)
\(二，设n为偶数，\)
\(则\left( \left( 2^{n\left( n+2\right)}-1\right)^{n+2}\right)^n+\left( \left( 2^{n\left( n+2\right)}-1\right)^{n+1}\right)^{n+1}=\left( \left( 2\times\left( 2^{n\left( n+2\right)}-1\right)\right)^n\right)^{n+2}\)
\(三，设n为大于等于2的正整数，\)
\(则\left( \left( 2^n-1\right)^n\right)^{n-2}+\left( \left( 2^n-1\right)^{n-1}\right)^{n-1}=\left( 2\times\left( 2^n一1\right)^{n-2}\right)^n\)

(第2题)
\(设\ x^a+y^b=z^c{,}\ 其中\ x{,}y{,}z{,}a{,}b{,}c{,}n为正整数，\)
\(则\left( xz^{nb}\right)^a+\left( yz^{na}\right)^b=z^{nab+c}\)

(第3题)
\(设\ x^a+y^b=z^c{,}\ 其中\ x{,}y{,}z{,}a{,}b{,}c{,}n{,}为正整数，\)
\(若a是nb的倍数，则\left( xz\right)^a+\left( yz^n\right)^b=z^{a+c}\)

(第4题)
\(设n为正整数，\)
\(则\left( 2^{2n+3}\right)^n+\left( 2^n\right)^{2n+3}=\left( 2^{n+1}\right)^{2n+1}=\left( 2^{2n+1}\right)^{n+1 }\)
\(则\left( 2^{2n-1}\right)^{n+1}+\left( 2^{n+1}\right)^{2n-1}=\left( 2^{2n+1}\right)^n=\left( 2^n\right)^{2n+1}\)


(第5题)
\(设n为正整数，n\left( \frac{n+1}{8}\right)=m，\)
\(则\left( 2^m\right)^{n-2}+\left( 2^{m-\left( \frac{n+1}{4}\right)}\right)^n=\left( 2^{m-\left( \frac{n-1}{2}\right)}\right)^{n+2}\)
\(注：其中n是\left( 8N-1\right)的数都有解\)

(第6题)
\(设n=a，n+1=b，a+b=c，\)
\(则\left( 2^{c+2}\right)^a+\left( 2^a\right)^{c+2}=\left( 2^b\right)^c=\left( 2^c\right)^{b}\)
\(则\left( 2^{c-2}\right)^b+\left( 2^b\right)^{c-2}=\left( 2^c\right)^a=\left( 2^a\right)^{c }\)

(第7题)
\(设n为大于等于0的正整数，\)
\(则\left( 2^n\right)^{n+2}+\left( 2^n\right)^{n+2}=\left( 2\times2^n\right)^{n+1}\)

(第8题)
\(设\ x^n+y^{n+1}=z^n,其中\ x{,}y{,}z{,}n{,}K为正整数，\)
\(且2^n-1=x=y{,}\ \ 2\left( 2^n-1\right)=z{,}\)
\(则\left( 2^n-1\right)^n+\left( 2^n-1\right)^{n+1}=\left( 2\left( 2^n-1\right)\right)^n\)
\(则\left( xK^{n+1}\right)^n+\left( yK^n\right)^{n+1}=\left( zK^{n+1}\right)^n\)
\(若n=ab，\)
\(则\left( \left( 2^{ab}-1\right)^b\right)^a+\left( 2^{ab\ }-1\right)^{ab+1}=\left( \left( 2\times\left( 2^{ab}-1\right)\right)^b\right)^a,\)
\(则\left( \left( 2^{ab}-1\right)^a\right)^b+\left( 2^{ab}-1\right)^{ab+1}=\left( \left( 2\times\left( 2^{ab}-1\right)\right)^a\right)^b\)
\(则\left( \left( 2^{ab}-1\right)^b\right)^a+\left( 2^{ab}-1\right)^{ab+1}=\left( \left( 2\times\left( 2^{ab}-1\right)\right)^a\right)^b\)
\(则\left( \left( 2^{ab\ \ }-1\right)^a\right)^b+\left( 2^{ab}-1\right)^{ab+1}=\left( \left( 2\times\left( 2^{ab}-1\right)\right)^b\right)^a
\)

(第9题)
\(设n为正整数，\)
\(则\left( 2^n\right)^{n+2}+\left( 2^n\right)^{n+2}=\left( 2\times2^n\right)^{n+1}\)
\(若n+2=ab{,}\ \ n+1=cd{,}\)
\(则\left( \left( 2^n\right)^a\right)^b+\left( \left( 2^n\right)^a\right)^b=\left( \left( 2\times2^n\right)^c\right)^d\)
\(则\left( \left( 2^n\right)^a\right)^b+\left( \left( 2^n\right)^a\right)^b=\left( \left( 2\times2^n\right)^d\right)^c\)
\(则\left( \left( 2^n\right)^b\right)^a+\left( \left( 2^n\right)^b\right)^a=\left( \left( 2\times2^n\right)^c\right)^d\)
\(则\left( \left( 2^n\right)^b\right)^a+\left( \left( 2^n\right)^b\right)^a=\left( \left( 2\times2^n\right)^d\right)^c\)
\(则\left( \left( 2^n\right)^a\right)^b+\left( \left( 2^n\right)^b\right)^a=\left( \left( 2\times2^n\right)^c\right)^d\)
\(则\left( \left( 2^n\right)^a\right)^b+\left( \left( 2^n\right)^b\right)^a=\left( \left( 2\times2^n\right)^d\right)^c\)

(第10题)
\(设x，n为正整数，\)
\(则2^{xn}+2^{xn}=2^{xn+1}\)
\(则\left( 2^n\right)^x+\left( 2^n\right)^x=2^{nx+1}\)
\(则\left( 2^x\right)^n+\left( 2^x\right)^n=2^{nx+1}\)
\(则\left( 2^n\right)^x+\left( 2^x\right)^n=2^{nx+1}\)
\(若mn=a，\)
\(则x^a+y^a=z^{a+1}\)

(第11题)
\(设m，n为奇数，\)
\(则\left( 2^m\right)^n+\left( 2^m\right)^n=\left( 2^{\left( mn+1\right)\div2}\right)^2\)
\(则\left( 2^m\right)^n+\left( 2^n\right)^m=\left( 2^{\left( mn+1\right)\div2}\right)^2\)
\(若mn=a，\)
\(则x^a+y^a=\left( z^{\left( a+1\right)\div2}\right)^2\)

(第12题)
\(设a，b，c，d，n都是奇数，\)其中
\(\left( abcd\cdots n\right)\div a=A，\)
\(\left( abcd\cdots n\right)\div b=B，\)
\(\left( abcd\cdots n\right)\div c=C，\)
\(\left( abcd\cdots n\right)\div n=N，\)
\(且方程左边项数等于x\)
\(则\left( x^A\right)^a+\left( x^B\right)^b+\cdots+\left( x^N\right)^n=\left( x^{\left( \left( ab\cdots n\right)+1\right)\div2}\right)^2\)

(第13题)
\(设x为大于等于2的正整数，n为任意正整数，x又为公式中的前项个数，\)
\(则x^n+x^n+\cdots+x^n=x^{(n+1)}{,}\ \ \ \ \ \ \ 简化公式：x(x^n)=x^{(n+1)}\)
\(x=2{,}代入公式得，2^n+2^n=2^{(n+1)，}，\)
\(x=3{,}代入公式得，3^n+3^n+3^n=3^{(n+1)}，\)
\(x=4{,}代入公式得，4^n+4^n+4^n+4^n=4^{(n+1)，}，\)
\(\cdots\cdots。\)

(第14题)
\(设n为任意奇数，\)
\(则2^n+2^n=\left\{ 2^{\left( n+1\right)\div2}\right\}^{^2}{,}\)
  \(2^1十2^1=2^2，\)
  \(2^3十2^3=4^2，\)
  \(2^5十2^5=8^2，\)
  \(2^7十2^7=16^2，\)
  \(......。\)

\((第15题)\)
\(设n为大于等于2的正整数，则\)
\(\left( 4^{\left( n+1\right)\times\left( n+2\right)\times n}\right)^{n-1}+\left( 4^{\left( n+1\right)\times\left( n+2\right)\times\left( n-1\right)}\right)^n\)\(+\)\(\left( 4^{\left( n\times\left( n+1\right)\times\left( n-1\right)+\left( n\times\left( n-1\right)\right)\right)}\right)^{n+1}+\)
\(\left( 4^{\left( n\times\left( n+1\right)\times\left( n-1\right)\right)}\right)^{n+2}=\left( 4^{\left( n\times\left( n+1\right)-1\right)}\right)^{\left( n\times\left( n+1\right)-1\right)}\)

\((第16题)\)


\((第17题)\)


\((第18题)\)
\(设x=(a+b+…+n)为大于等于3的奇数，\)
\((a^2+b^2+\cdots+n^2)为y，其中abn均为正整数，\)
\(则a^2+b^2+\cdots+n^2+\left\{ \frac{(x^2-1)}{2}-\frac{(x^2-y)}{2}\right\}^2=\left\{ \frac{(x^2+1)}{2}-\frac{(x^2-y)}{2}\right\}^2\)
\(实例：x=5{,}\ \ \ \ \ 5^2+12^2=13^2{,}\)
\(5=1+1+1+1+1{,}代入公式得1^2+1^2+1^2+1^2+1^2+2^2=3^2，\)
\(5=1+1+1+2{,}代入公式得1^2+1^2+1^2+2^2+3^2=4^2，\)
\(5=1+1+3{,}代入公式得1^2+1^2+3^2+5^2=6^2，\)
\(5=1+2+2{,}代入公式得1^2+2^2+2^2+4^2=5^2，\)
\(5=1+4{,}代入公式得1^2+4^2+8^2=9^2，\)
\(5=2+3{,}代入公式得2^2+3^2+6^2=7^2，\)
简化公式：
\(设(a^2+b^2+\cdots+n^2)=x{,}其中(a+b+\cdots+n)为大于等于3的奇数，且a，b，n，x为正整数，\)
\(则a^2+b^2+\cdots+n^2+\left\{ \frac{(x-1)}{2}\right\}^2=\left\{ \frac{(x+1)}{2}\right\}^2\)  

\((第19题)\)
\(设n为正整数，\)
\(则\left( 2^{n+2}\right)^n+\left( 2^n\right)^{n+2}=\left( 2^{n+1}\right)^{n+1}\)
\(则\left( 2^{n+2}\right)^n+\left( 2^{n+2}\right)^n=\left( 2^{n+1}\right)^{n+1}\)
\(则\left( 2^n\right)^{n+2}+\left( 2^n\right)^{n+2}=\left( 2^{n+1}\right)^{n+1}\)
\(若n\left( n+2\right)=ab{,}\ \ \ \left( n+1\right)\left( n+1\right)=cd{,}\)
\(则\left( 2^a\right)^b+\left( 2^b\right)^a=\left( 2^c\right)^d=\left( 2^d\right)^c\)
\(则\left( 2^a\right)^b+\left( 2^a\right)^b=\left( 2^c\right)^d=\left( 2^d\right)^c\)
\(则\left( 2^b\right)^a+\left( 2^b\right)^a=\left( 2^c\right)^d=\left( 2^d\right)^c\)

cz1

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cz1

lusishun

求出一组
X^5+y^3=z^7
的最小正整数解。对您来说，小菜一碟

lusishun

lusishun 发表于 2024-2-17 17:31
求出一组
X^5+y^3=z^7
的最小正整数解。对您来说，小菜一碟

(2^18)^5+(2^30)^3=(2^13)^7

朱明君

\(\left( 2^{30}\right)^3+\left( 2^{18}\right)^5=\left( 2^{13}\right)^7\)
\(\left( 2^7\right)^5+\left( 2^5\right)^7=\left( 2^4\right)^9，\ \ \ \ \ \left( 2^{70}\right)^5+\left( 2^{50}\right)^7=\left( 2^{39}\right)^9\)
\(\left( 2^{36}\right)^7+\left( 2^{28}\right)^9=\left( 2^{23}\right)^{11}\)
\(\left( 2^{88}\right)^9+\left( 2^{72}\right)^{11}=\left( 2^{61}\right)^{13}\)
\(\left( 2^{169}\right)^{11}+\left( 2^{143}\right)^{13}=\left( 2^{124}\right)^{15}\)
\(\left( 2^{30}\right)^{13}+\left( 2^{26}\right)^{15}=\left( 2^{23}\right)^{17}\)
......。

朱明君

\(\left( 2^{119}\right)^{15}+\left( 2^{105}\right)^{17}=\left( 2^{94}\right)^{19}\)
\(\left( 2^{247}\right)^{17}+\left( 2^{221}\right)^{19}=\left( 2^{200}\right)^{21}\)

\(\left( 2^{69}\right)^{21}+\left( 2^{63}\right)^{23}=\left( 2^{58}\right)^{25}\)

朱明君

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