Question: Suppose a, a+d, a+2d, ..., a+(n-1)d are primes. Find the largest possible n, under the condition that d cannot be a mulitple of 10.
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(Table of primes up to 1000)Answer: Lets make a guess. We claim the largest possible n≥5.
a must be odd, otherwise a+2d is an even number greater than 2 and thus not a prime.
If d is odd, then a+d is an even prime greater than 2 and thus not a prime.
If d is even and not a multiple of 10, then one of a, a+d, a+2d, a+3d, a+4d have the last digit being 5 and hence it must be 5. Note that there is only one odd prime, i.e. 3, which is smaller than 5. Therefore it is only possible if a=5 or a+d=5.
If a+d=5 then a=3, d=2. The sequence is just 3,5,7.
The only case left is that a=5 and then a+5d is a multiple of 5. Therefore n is at most 5.
You can try to search using the table.
5, 11, 17, 23, 29 is a prime sequence with d=6.
5, 17, 29, 41, 53 is a prime sequence with d=12.
Therefore the largest possible n=5.
What if d is allowed to be a multiple of 10?
No matter how large n is, we can construct a sequence a, a+d, a+2d, ..., a+(n-1)d of primes. It is the famous Green-Tao theorem, proved by Ben Green and Terence Tao in 2004.
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