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Challenge 360: Prime Sequence

One of Dr Reasoner's students has been investigating prime numbers, and was very excited to notice that the terms of the sequence starting at 41, with differences increasing by the next even number each time (41, 43, 47, 53, 61.... ) all seem to be prime numbers! The student thinks this is a fine way to find new primes.

Is the student correct that all the terms in this sequence are prime? Find a convincing argument to support your answer. Your argument should not rely on the use of technology!