So is 1!−2!+3!=5 1!−2!+3!−4!+5!=101 1!−2!+3!−4!+5!−6!+7!=4421 So, I guess the question is how often is this sum prime vs. not.
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... - 8! + 9! = 326981 = 79 * 4139, not prime. But you could still ask about frequency.
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Today is 139th day of year 2020 139 = 9×8+7×6+5×4+3×2-1
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Does this pattern continue?
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I don't have proof, but pretty sure it doesn't. The numbers of this form are getting larger and larger, so you'd get a sequence that generates infinitely many primes, which would be one of the holy grails of number theory.
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Prove it!

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Rey’s number is sum of infinite series of factorials R = 10^-x1! + 10^-x2! + 10^-x3! +… Where x = random positive integer x1 < x2 < x3… 0 < R < 1 Rey’s number is transcendental and uncomputable
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Now turn your calculator upside down and see what it says.
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indeed
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