Brainteaser 1659: Prime cuts

From The Sunday Times, 26th June 1994 [link]

My new season ticket number is a prime obtained by rearranging five consecutive digits. By cutting off successive single digits from alternate ends I can steadily reduce this to a four-figure prime, a three-figure prime, a two-figure prime and finally a one-figure prime.

You too can use some short cuts to find my season ticket number.

This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant).