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The Green-Tao theorem asserts that there are arithmetic progressions of prime numbers that are arbitrarily long, 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 $$2,166,703,103,992,332,274,919,550$$, 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 $$2^{2^{2^{2^{2^{2^{2^{100n}}}}}}}$$ (E(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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