The term "mod" stands for "modulo" and is used in mathematics to indicate the remainder after dividing one number by another.
The "mod 5^3" indicates that the equation is being solved in the ring of integers modulo 5^3, which means that all calculations are done using only the numbers 0 to 5^3-1.
There are multiple solutions to this equation, as it is a quadratic equation. In the ring of integers modulo 5^3, the solutions are 2 and 123.
This equation is an example of a congruence relation, which is a fundamental concept in modular arithmetic. It states that X^2 + 1 is congruent to 0 (or equivalent to 0) when considered in the ring of integers modulo 5^3.
No, this equation cannot be solved using traditional algebraic methods because it involves modular arithmetic, which has its own set of rules and properties. However, the equation can be solved using specialized techniques in modular arithmetic.