Mini-challenge: First Prime of form 403333...

[Ron Ross]

"It is not immediately obvious, but not all the terms of this sequence are composite. There are in fact an infinite number of primes in this sequence. However, the smallest one is surprisingly large." Quote of Joe Horn.
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Just out of curiosity, how can you make this statement? Is this a countable sequence of sorts or of a type that you know will always produce one more prime? I am a novice so please, keep your explanation simple if possible.
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If I remember correctly, the Mersenne Primes are of such rarity that they are yet to be determined to be infinite or not. Will your prime generator always produce, or will it come up empty at some extremely large value. Just ask'en.