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goosethe@lemmy.sdf.org

Is there an interesting set of natural numbers defined by a number-theoretic property that is finite? (lemmy.sdf.org)

submitted 1 year ago* (last edited 1 year ago) by CanadaPlus@lemmy.sdf.org to c/math@lemmy.sdf.org

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[–] OscarCunningham@lemmy.world 2 points 8 months ago (1 children)

The Heegner Numbers. These are the n such that ℚ[√-n] has unique factorisation. There are exactly 9 of them:

1, 2, 3, 7, 11, 19, 43, 67, 163.

A famous fact about them is that 163 being a Heegner Number leads to e^(π√163) being very close to a whole number.

262537412640768743.99999999999925…

[–] CanadaPlus@lemmy.sdf.org 1 points 8 months ago

TIL about prime-generating quadratic polynomials, as well. I feel like I'm destined to use one in code now. The logic behind e^π√163^ looks like more than I can absorb today, haha.

Because I find Wikipedia doesn't explain it in the best way, a quadratic field like ℚ[√-n] is literally just the field of rationals with √-n and all the new numbers you can make with it added.