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The Green-Tao theorem asserts that there are arithmetic progressions of prime numbers that are arbitrarily long[1], and some of the common differences of these sequences can be large even if the sequence is short. For example, the smallest common difference of the first 19-term arithmetic sequence of prime numbers is $$2,166,703,103,992,332,274,919,550$$[2], or about 2 septillion using the short scale.

Even larger numbers come from a weak upper bound given by Terence Tao that the first arithmetic progression of length $$n$$ of primes has all of its entries less than $$2^{2^{2^{2^{2^{2^{2^{100n}}}}}}}$$[3] (E[2](100n)#7 using Hyper-E Notation).

## Approximations

19-term sequence number:

Notation Approximation
Scientific notation $$2.167\times 10^{24}$$

Upper bound for entries in 100-long sequence:

Notation Approximation
Arrow notation $$2\uparrow 2\uparrow 2\uparrow 2\uparrow 2\uparrow 2\uparrow 2\uparrow 10\uparrow 4$$ (exact)
$$\approx 2\uparrow\uparrow 11$$
BEAF $$\{2,\{2,\{2,\{2,\{2,\{2,\{2,10000\}\}\}\}\}\}\}$$ (exact)
$$\approx\{2,11,2\}$$

## Sources

1. B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions (2004, accessed 2020-11-10)
2. OEIS, sequence A061558 (Accessed 2020-11-10)
3. UCLA, Terence Tao: Structure and Randomness in the Prime Numbers (2009, accessed 2020-11-10)
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