Is there an interesting set of natural numbers defined by a number-theoretic property that is finite?
Is there an interesting set of natural numbers defined by a number-theoretic property that is finite?
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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…
TIL about prime-generating quadratic polynomials, as well. I feel like I'm destined to use one in code now. The logic behind eπ√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.