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.

Pyth, 18 15 14 bytes

Thanks to @Maltysen for -1 byte

#=-QefP_TUQ)q1

A program that takes input on STDIN and prints True or False as appropriate.

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How it works

#=-QefP_TUQ)q1  Program. Input: Q
#          )    Loop until error statement (which occurs when Q<3):
         UQ      Yield [0, 1, 2, 3, ..., Q-1]
     fP_T        Filter that by primality
    e            Yield the last element of that
 =-Q             Q = Q - that
            q1  Q is 1 (implicit variable fill)
                Implicitly print

Old version with reduce, 18 bytes

qu-G*<HGH_fP_TSQQ1

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How it works

qu-G*<HGH_fP_TSQQ1  Program. Input: Q
              SQ    Yield [1, 2, 3, ..., Q]
          fP_T      Filter that by primality
         _          Reverse it
 u                  Reduce it:
                Q    with base case Q and
                     function G, H -> 
     <HG              H<G
    *   H             *H (yields H if H<G, else 0)
  -G                  Subtract that from G
q                1  The result of that is 1
                    Implicitly print

Jelly, 9 8 bytes

’ÆRṪạµ¡Ḃ

Try it online! or verify all test cases.

How it works

’ÆRṪạµ¡Ḃ  Main link. Argument: n

     µ    Combine all atoms to the left into a chain.
’           Decrement; yield n - 1.
 ÆR         Prime range; yield all primes in [2, ..., n -1].
   Ṫ        Tail; yield p, the last prime in the range.
            If the range is empty, this yields p = 0.
    ạ       Compute the absolute difference of p and n.
      ¡   Call the chain to the left n times.
          This suffices since each iteration decreases n, until one of the fixed
          points (1 or 2) is reached.
       Ḃ  Bit; return the parity of the fixed point.