npub14lmh8azph6ll8fsf6xx4rc8882ulaaagsautxufnv0p7w3qedrhsa75crk (npub14lm…5crk) a) The factorial isn't defined for negative integers. For x ≥ 0 we have a solution at x = 0, since 0! and 2⁰ are both 1. At x = 1 and x = 2 we have no solution since 1! < 2¹ and 2! < 2². For larger x, x! has a factor of 3, and so cannot be equal to 2ˣ. So there is one solution total.
b) When the factorial is extended to the reals, it has an asymptote at every negative integer. Between these asymptotes it gets close to 0. In fact near -(n+1/2) the value of x! is approximately ±1/n!. Since this is shrinking much faster than 2ˣ, the two functions cross near every asymptote, meaning that this equation has infinitely many solutions.
Oscar Cunningham /
npub1my…dtj8l
2023-08-21 10:20:50
in reply to nevent1q…03sd
Author Public Key
npub1my7d7x8gs4a96erlhhj9hn2lstz9qrdhex4hhuhgq9kqe5z06k6qldtj8lPublished at
2023-08-21 10:20:50Event JSON
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"content": "nostr:npub14lmh8azph6ll8fsf6xx4rc8882ulaaagsautxufnv0p7w3qedrhsa75crk a) The factorial isn't defined for negative integers. For x ≥ 0 we have a solution at x = 0, since 0! and 2⁰ are both 1. At x = 1 and x = 2 we have no solution since 1! \u003c 2¹ and 2! \u003c 2². For larger x, x! has a factor of 3, and so cannot be equal to 2ˣ. So there is one solution total.\n\nb) When the factorial is extended to the reals, it has an asymptote at every negative integer. Between these asymptotes it gets close to 0. In fact near -(n+1/2) the value of x! is approximately ±1/n!. Since this is shrinking much faster than 2ˣ, the two functions cross near every asymptote, meaning that this equation has infinitely many solutions.",
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