Number Theory as a Service

Linear Diophantine Equations

I'm interested in the equation

x

+

y

=

.

I do not wish to guess what the answer is.

I think there are no solutions.

I don't know the general solution, but I think that one solution is given by

x

=

,

y

=

.

I think that a general solution is given by

x	=		+	k;
y	=		+	k;	(where k∈Z)

I think that any x and any y would work.

Simplifying Congruences

I'm interested in the congruence

x

≡

(mod ).

I do not wish to guess what the answer is.

I think there are no such numbers x.

I think the condition is equivalent to saying

x

≡

(mod ).

Simultaneous Congruences

I'm interested in the simultaneous congruences

x	≡		(mod );
x	≡		(mod ).

I do not wish to guess what the answer is.

I think there are no solutions.

I think the condition is equivalent to saying

x

≡

(mod ).