Crazy Diophantus

Number Theory Level pending

How many integral solutions are there for the following equation?

x 12 + 4 x 9 + 6 x 6 + 8 x 3 + 6 = y 4 { x }^{ 12 }+4{ x }^{ 9 }+6{ x }^{ 6 }+8{ x }^{ 3 }+6={ y }^{ 4 }


The answer is 2.

Ραμών Αδάλια by Ραμών Αδάλια , 22, Spain

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1 solution

x 12 + 4 x 9 + 6 x 6 + 8 x 3 + 6 = y 4 x^{12} + 4x^{9} + 6x^{6} + 8x^{3} + 6 = y^{4}

x 4 ( x 8 + 4 x 5 + 6 x 2 ) + 8 x 3 + 6 = y 4 x^{4}(x^{8} + 4x^{5} + 6x^{2}) + 8x^{3} + 6 = y^{4}

x 4 ( x 8 + 4 x 5 + 6 x 2 ) ≡ 0 , 1 ( m o d 16 ) . x^{4}(x^{8} + 4x^{5} + 6x^{2}) \equiv 0,1\pmod{16}.

8 x 3 ≡ 0 , 8 ( m o d 16 ) . 8x^{3} \equiv 0,8\pmod{16}.

y 4 ≡ 0 , 1 ( m o d 16 ) . y^{4} \equiv 0,1\pmod{16}.

W h e n When 2 ∣ x 2\mid x

0 + 0 + 6 ≡ 0 , 1 ( m o d 16 ) . 0 + 0 + 6 \equiv 0,1\pmod{16}.

W h e n When 2 ∤ x 2\nmid x

1 + 8 + 6 ≡ 0 , 1 ( m o d 16 ) . 1 + 8 + 6 \equiv 0,1\pmod{16}.

Any case meets therefore no solutions.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Completely unexpected solution! I did not do it this way, yours is much more intuitive!

Ραμών Αδάλια - 5 years, 1 month ago

I'm not sure what you mean by "Any case meets therefore no solution".

There is a solution of ( − 1 , 1 ) (-1, 1) , so there is at least one solution.

Calvin Lin Staff - 5 years ago

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