One Hundred Is Not Squarefree Plus a Power of Two

Every integer $N \\ge 2$ is a squarefree number plus a power of two: the assertion is false, and the smallest witness is $100$. We give the arithmetic progression $1764t + 100$, no member of which admits such a representation, and show that its members have positive lower density among the counterexamples below any bound. The obstruction is a covering argument in three congruences, carried by the single identity $1764 = \\operatorname{lcm}(9, 49, 4)$: the three ranges of the exponent are handled by the three square divisors $3^2$, $7^2$ and $2^2$, and no exponent escapes all three. Every statement is machine-checked.