Let $d$ be day, $m$ be month and $y$ be year fields of a date. I want to find few dates of format $$(d^2\, mod\,\, 2 + (my + d^3) \,mod \,4) = 2$$

Is there a method to solve this type of equation or tell whether there exists a solution for this type of equation?

Here is my try: I framed this problem as solving constrained multivariable diophantine equations.

$$(d^2 - 2q_1) + (my + d^3 - 4q_2) = 2 $$ Here $q_1$ is the quotient when divided by 2 and $q_2$ is the quotient when divided by 4

subject to:

$0 <= d^2 - 2q_i <= 1$ //remainder when divided by 2 is between 0 and 1

$0 <=my + d^3 - 4q_2 <= 3$ //remainder when divided by 4 is between 0 and 3

$1 <= d <= 31$ //date value is always between 1 and 31

$1 <= m <= 12$ //month value is always between 1 and 12

$y > 0 $ //year has to be a positive value

I planned to solve (how I am going to do I don't know) the above problem and obtain d, m and y values and check whether it is a proper date(leap year constraints and few months have only 30 days). What is the method to do so? Is it at least possible to say whether a feasible solution exists for problems of above format?


Note: I am answering my own question

According to Hilbert's tenth problem, there exists no general algorithm to say whether there exist integer solutions for a given Diophantine equation or not.


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