prime(n) is the n-th prime, and valuation(n, p) the p-adic valuation - #1453
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A function of its own -- PP carries no order -- with SymPy's spelling: prime(1) is 2, prime(25) is 97, NaN off the positive whole numbers, and left as written for a symbol or an index past a million, where the table would be built towards it. Its codomain is Domain.Prime. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
The exponent of the prime p in n: valuation(12, 2) is 2, valuation(12, 5) is 0, +oo at n = 0 by the convention, NaN where p is not prime. Sullivan and Mackey's Ex 5.5.1 writes n = 2^m (2 l + 1) with m the 2-adic valuation, and n = product(p^valuation(n, p), p in PP) is what a factorisation says over the set of primes. v_p(n) in LaTeX, sympy.multiplicity in SymPy. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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The two functions #1450's questions come down to, beside the set #1452 added.
prime(n)— then-th prime, SymPy's spelling:prime(1)is2,prime(25)is97,prime(1000)is7919;NaNoff the positive whole numbers; left as written for a symbol or an index past a million, where the table would be built towards it. A function of its own rather than an operation onPP, which carries no order; its codomain isDomain.Prime. The table behind it,Primes.GetPrime, existed already for the continued-fraction code — this exposes it.\operatorname{prime}(n)in LaTeX,sympy.prime(n).valuation(n, p)— thep-adic valuation, the exponent of the primepinn:valuation(12, 2)is2,valuation(12, 5)is0,valuation(-40, 2)is3,+ooatn = 0by the convention every text uses,NaNwherepis not prime. Sullivan and Mackey's Ex 5.5.1 writesn = 2^m (2 l + 1)withmthe 2-adic valuation, andn = product(p^valuation(n, p), p in PP)is what a factorisation says over the set — theproduct(f, p in S)binder itself is the multiplicative twin of #1435'sunionand waits for #1019's indexed-family convention.v_p(n)in LaTeX,sympy.multiplicity(p, n).Both are nodes through every pipeline (
WritingARule.md's buildable count goes 48 → 50), with 22 rows inPrimeSetTest,Syntax.mdandBREAKING-CHANGES.md. Full suite 12499 passed; the allocation gate passes on all 19 gated benchmarks —ParseHard+0.7%, which is #1441's ~12 KB per'name('token, twice.Part of #1450.
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