Can the number reach 1 by repeatedly subtracting the largest prime less than it?
Retina, 31 bytes

.+
$*
+`1(?!(11+)\1+$)11+
1
^1$


Prints 0 (falsy) or 1 (truthy).

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Explanation

.+
$*


Convert input to unary by turning input N into N copies of 1.

+`1(?!(11+)\1+$)11+
1


Repeatedly remove the largest prime less than the input. This is based on the standard primality test with regex.

^1$


Check whether the result is a single 1.