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Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ cannot be deduced from the $p$-adic method. See my answer to Diophantine equation with no integer solutions, but with solutions modulo every integer for more details.

Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ canno be deduced from the $p$-adic method.

Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ cannot be deduced from the $p$-adic method.

Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ cannot be deduced from the $p$-adic method. See my answer to Diophantine equation with no integer solutions, but with solutions modulo every integer for more details.

Yes, there are similar modulos to other powers. There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

Yes, there are similar modulos to other powers. For example, $n^6\mod504$ can take only the values $0, 1, 64, 217, 225, 280, 288, 441$ over integers $n$.

The $p$-adic methods are used to prove the insolubility for some Diophantine equations. For example:

There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.

Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.

However, the insolubility to the negative Pell equation $x^2-34y^2=1$ cannot be deduced from the $p$-adic method.