不定方程x^a+y^b=z^c的研究

朱明君

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

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

\((第1题公式一)\)
X^17+y^18=z^19
\(\frac{17\times19+1}{2}=162，\frac{18}{2}=9，\)
\(\left( 2^{162}\right)^{17}+\left( 2^{\left( 162-9\right)}\right)^{18}=\left( 2^{\left( 162-17\right)}\right)^{19}\)

\((第2题公式一)\)
x^2023+y^2024=z^2025
\(\frac{2023\times2025+1}{2}=2048288{,}\ \ \ \ \frac{2024}{2}=1012\)
\(\left( 2^{2048288}\right)^{2023}+\left( 2^{\left( 2048288-1012\right)}\right)^{2024}=\left( 2^{\left( 2048288-2023\right)}\right)^{2025}\)

\((第3题公式二)，n=2,\)
\(\left( \left( 2^8-1\right)^4\right)^2+\left( \left( 2^8-1\right)^3\right)^3=\left( 2\times\left( 2^8-1\right)^2\right)^4\)

\((第4题公式二)，n=10,\)
\(\left( \left( 2^{120}-1\right)^{12}\right)^{10}+\left( \left( 2^{120}-1\right)^{11}\right)^{11}=\left( 2\times\left( 2^{120}-1\right)^{10}\right)^{12}\)

\((第5题公式三)，n=2,\)

\(\cdots\cdots。\)





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


\((第2题)\)


\((第3题)\)


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

\((第5题)\)
\(设n为正整数，\)
\(则\left( 2^{n+2}\right)^n+\left( 2^n\right)^{n+2}=\left( 2^{n+1}\right)^{n+1}\)
\(则\left( 2^{n+2}\right)^n+\left( 2^{n+2}\right)^n=\left( 2^{n+1}\right)^{n+1}\)
\(则\left( 2^n\right)^{n+2}+\left( 2^n\right)^{n+2}=\left( 2^{n+1}\right)^{n+1}\)
\(若n\left( n+2\right)=ab{,}\ \ \ \left( n+1\right)\left( n+1\right)=cd{,}\)
\(则\left( 2^a\right)^b+\left( 2^b\right)^a=\left( 2^c\right)^d=\left( 2^d\right)^c\)
\(则\left( 2^a\right)^b+\left( 2^a\right)^b=\left( 2^c\right)^d=\left( 2^d\right)^c\)
\(则\left( 2^b\right)^a+\left( 2^b\right)^a=\left( 2^c\right)^d=\left( 2^d\right)^c\)



(第1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2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3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4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5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6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7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8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9题)
\(设n为任意奇数，\)
\(则2^n+2^n=\left\{ 2^{\left( n+1\right)\div2}\right\}^{^2}{,}\)
  \(2^1十2^1=2^2，\)
  \(2^3十2^3=4^2，\)
  \(2^5十2^5=8^2，\)
  \(2^7十2^7=16^2，\)
  \(......。\)

朱明君

，，，，，，，，

朱明君

,,,,,,,,,,,,,,,,,,,,,,,,,,

朱明君

，，，，，，，，