How is the Green-Tao theorem true So let's say I started my sequence with a number that ends in a three with an even common difference because an odd common diff would give us an even, non prime number. If I add on a 2 repeatedly then that means that I will have a sequence of 3 then a number that ends in 5. So obviously there can't be a 2 term sequence that starts with 3 and has a common difference of 2.
If I try out 3 (the starting number) with a common diff (cd from now on) of 4 then I get 3, number ending in 7, number ending in 1, number ending in 5. So this sequence can only have 3 primes.
If I try out 3 with a common diff of 6 then I get 3, 9, then 5. Again since the number ends in 5, then the sequence can only have 2 primes.
So I did this for 3,5,7 and 9 and found that all of them ended in 5 at one point or another.
So my question is how can there be a 1,000,000 term sequence of primes when after I tested out the possible last digits of sequences that start with primes, they all end in a 5 at some point before the 1,000,000th term.
 A: The problem with your argument is that you don't consider large successive differences; while it is easy to find an upper bound on the possible lengths of sequences with successive difference at most $k$, that upper bound grows with $k$, so this doesn't prevent arbitrarily long arithmetic sequences from appearing (since the "longer" sequences might have "larger" successive differences).
For instance, what if we begin with a prime number with last digit $3$, and our successive difference is $30$? Then we never hit a multiple of $2, 3$, or $5$. Larger and larger successive differences let us in principle get longer and longer arithmetic sequences, so your style of argument is not an obstacle to Green-Tao.
And in fact we can find examples of long progressions with large differences; e.g. here the example $$199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089$$ is given, which is ten terms long (with successive difference $210$). Although longer arithmetic progressions take increasingly long to find, they do exist.
