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The Prouhet–Tarry–Escott problem is a classic problem in number theory and combinatorial mathematics. It is named after three mathematicians: Pierre Prouhet, Édouard Tarry, and John Escott. The problem can be stated as follows: Given a set of integers, the goal is to find a way to partition these integers into two groups such that the sums of the integers in each group are equal.
The Ramanujan–Nagell equation is a famous equation in number theory given by: \[ 2^n = x^2 + 7 \] where \(n\) is a non-negative integer and \(x\) is an integer. The equation states that \(2^n\) can be expressed as the sum of a perfect square \(x^2\) and the integer 7. The equation is considered particularly interesting because it leads to a list of specific solutions.