兔年快乐

王守恩

今年是兔年，恭祝大家兔年快乐！

F=0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765,

\(\ x^{F_{n}}+y^{F_{n+1}}=z^{F_{n+2}}\)

\(\ x^{F_{n}}+y^{F_{n+1}}=z^{F_{n+2}}\)

\(\ x^{F_{n}}+y^{F_{n+1}}=z^{F_{n+2}}\)

\(\ x^{F_{n}}+y^{F_{n+1}}=z^{F_{n+2}}\)

\(\ x^{F_{n}}+y^{F_{n+1}}=z^{F_{n+2}}\)

drc2000再来

你也兔年快乐！
菲波拉切数列,每项都为前2项之和.

上述识子是指数形式么?请说明.

ysr

1^0+1^1=1^1
2^1+2^1=2^2
4^1+2^2=2^3
（√24）^2+2^3=2^5
(√27)^2+6^3=3^5

这个行吗？

王守恩

今年是兔年，恭祝大家兔年快乐！ a=2, 3, 4, 5, 6, 7, 8, 9, ......

F=0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765,

\(\big((a^{01}-1)^{01}\big)^{00}+\big((a^{01}-1)^{01}\big)^{01}=\big(a(a^{01}-1)^{00}\big)^{01}\)
\(\big((a^{02}-1)^{03}\big)^{01}+\big((a^{02}-1)^{02}\big)^{01}=\big(a(a^{02}-1)^{01}\big)^{02}\)
\(\big((a^{03}-1)^{03}\big)^{01}+\big((a^{03}-1)^{02}\big)^{02}=\big(a(a^{03}-1)^{01}\big)^{03}\)
\(\big((a^{05}-1)^{08}\big)^{02}+\big((a^{05}-1)^{05}\big)^{03}=\big(a(a^{05}-1)^{03}\big)^{05}\)
\(\big((a^{08}-1)^{08}\big)^{03}+\big((a^{08}-1)^{05}\big)^{05}=\big(a(a^{08}-1)^{03}\big)^{08}\)
\(\big((a^{13}-1)^{21}\big)^{05}+\big((a^{13}-1)^{13}\big)^{08}=\big(a(a^{13}-1)^{08}\big)^{13}\)
\(\big((a^{21}-1)^{21}\big)^{08}+\big((a^{21}-1)^{13}\big)^{13}=\big(a(a^{21}-1)^{08}\big)^{21}\)
\(\big((a^{34}-1)^{55}\big)^{13}+\big((a^{34}-1)^{34}\big)^{21}=\big(a(a^{34}-1)^{21}\big)^{34}\)
\(\big((a^{55}-1)^{55}\big)^{21}+\big((a^{55}-1)^{34}\big)^{34}=\big(a(a^{55}-1)^{21}\big)^{55}\)
......

王守恩

今年是兔年，恭祝大家兔年快乐！  a=2, 3, 4, 5, 6, 7, 8, 9, ......

Pell numbers: 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, ....

\(\big((a^{002}-1)^{002}\big)^{00}+\big((a^{002}-1)^{001}\big)^{001}=\big(a(a^{002}-1)^{00}\big)^{002}\)
\(\big((a^{005}-1)^{005(002-1)}\big)^{001}+\big((a^{005}-1)^{002*001+1}\big)^{002}=\big(a(a^{005}-1)^{001(002-1)}\big)^{005}\)
\(\big((a^{012}-1)^{012}\big)^{02}+\big((a^{012}-1)^{005}\big)^{005}=\big(a(a^{012}-1)^{02}\big)^{012}\)
\(\big((a^{029}-1)^{029(012-1)}\big)^{005}+\big((a^{029}-1)^{012*011+1}\big)^{012}=\big(a(a^{029}-1)^{005(012-1)}\big)^{029}\)
\(\big((a^{070}-1)^{070}\big)^{12}+\big((a^{070}-1)^{029}\big)^{029}=\big(a(a^{070}-1)^{12}\big)^{070}\)
\(\big((a^{169}-1)^{169(070-1)}\big)^{029}+\big((a^{169}-1)^{070*069+1}\big)^{070}=\big(a(a^{169}-1)^{029(070-1)}\big)^{169}\)
\(\big((a^{408}-1)^{408}\big)^{70}+\big((a^{408}-1)^{169}\big)^{169}=\big(a(a^{408}-1)^{70}\big)^{408}\)
\(\big((a^{985}-1)^{985(408-1)}\big)^{169}+\big((a^{985}-1)^{408*407+1}\big)^{408}=\big(a(a^{985}-1)^{169(408-1)}\big)^{985}\)
............

dodonaomikiki

王老师，辛苦啦！


一直帮助我，解决那我的很多迷惑！
祝愿老师在新年礼，圣体健康，快快乐乐

王守恩

“1” 是数学殿堂里最活跃的数。\(1+2^3=3^2\)

\((2n^3)^{6}+(2n^2)^{9}=2^6n^{18}+2^{9}n^{18}=(2^{6}n^{18})(1+2^3)=(2^{6}n^{18})(3^2)=(24n^9)^2\)

\((2×1^3)^{6}+(2×1^2)^{9}=(24×1^9)^{2}\)
\((2×2^3)^{6}+(2×2^2)^{9}=(24×2^9)^{2}\)
\((2×3^3)^{6}+(2×3^2)^{9}=(24×3^9)^{2}\)
\((2×4^3)^{6}+(2×4^2)^{9}=(24×4^9)^{2}\)
\((2×5^3)^{6}+(2×5^2)^{9}=(24×5^9)^{2}\)
\((2×6^3)^{6}+(2×6^2)^{9}=(24×6^9)^{2}\)
\((2×7^3)^{6}+(2×7^2)^{9}=(24×7^9)^{2}\)
\((2×8^3)^{6}+(2×8^2)^{9}=(24×8^9)^{2}\)
\((2×9^3)^{6}+(2×9^2)^{9}=(24×9^9)^{2}\)

cz1

\(1^2+2^2+3^2+......+24^2=70^2\)

\(18^2+19^2+20^2+......+28^2=77^2\)

\(25^2+26^2+27^2+......+50^2=195^2\)

\(38^2+39^2+40^2+......+48^2=143^2\)

\(456^2+457^2+458^2+......+466^2=1529^2\)

\(854^2+855^2+856^2+......+864^2=2849^2\)


通过以上数据，朱火华也导不出通解公式！

冒牌数学家：朱火华：冒牌数学家

王守恩

简单才是最好！这是最小解。

\((2^2)^{0}+(2^0)^{2}=(2^1)^{1}\)
\((2^3)^{1}+(2^1)^{3}=(2^2)^{2}\)
\((2^4)^{2}+(2^2)^{4}=(2^3)^{3}\)
\((2^5)^{3}+(2^3)^{5}=(2^4)^{4}\)
\((2^6)^{4}+(2^4)^{6}=(2^5)^{5}\)
\((2^7)^{5}+(2^5)^{7}=(2^6)^{6}\)
\((2^8)^{6}+(2^6)^{8}=(2^7)^{7}\)
\((2^9)^{7}+(2^7)^{9}=(2^8)^{8}\)

Treenewbee

\[3^1+1^3=2^2\]