A Unique Characterization of the Number 17 via Digit-Sum Square Conditions

Abstract

We prove that the number 17 is the unique integer n ≥ 10 satisfying the simultaneous conditions: n - s(n) = a², n + s(n) = b², where s(n) denotes the sum of the decimal digits of n and a, b are non-negative integers. We further show that adding the condition √(n-1) ∈ ℤ preserves uniqueness. We also prove that the identity: √(n-1) = (√(n-s(n)) + √(n+s(n))) / 2 holds for n ≥ 10 if and only if n = 17. Finally, we introduce a generative system S₁₇ that produces the integers from 1 to 10 using only 17 and permitted operations. Keywords: Number Theory, Digit Sum, Square Numbers, Number 17, Fermat Prime.