Extracting 3-adic factors of signed integers

Goal and scope

v3(-108)=3,v3(18)=2,v3(-1944)=5,v3(-90)=2; zero uses infinity.

Exact extraction at prime 3, product/sum instances and an unequal-sum argument; no completion.

Givens and symbols

• calc

• G1 — p=3 is prime, a=-108,b=18; v_p(0) uses the sole serializable string sentinel infinity.

• u,v — unit parts after extraction, possibly negative
• v_p(0) — infinity string

Rules and sources

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Assume unique prime factorization and that a prime not dividing either factor does not divide their product.
• mko_definition · cited — Require prime p≥2. For a≠0 uniquely write |a|=p^k*u with p not dividing u, and define v_p(a)=k. Negative integers are valid and normalized by absolute value. Use the single serializable infinity sentinel for v_p(0). Exact families check factor extraction, v_p(ab)=v_p(a)+v_p(b), and v_p(a+b)≥min(v_p(a),v_p(b)). Unequal summand valuations guarantee equality in the sum inequality; equal valuations do not guarantee it and may cancel to zero. Composite bases are outside the p-adic valuation convention. Identities involving zero explicitly use the infinity convention. · mko-p-adic-valuation-integers

Main derivation

p=3,\quad a=-108

|a|=108=3^3\cdot4,\quad3\nmid4\Rightarrow v_3(a)=3

b=18

18=3^2\cdot2,\quad3\nmid2\Rightarrow v_3(b)=2

ab=-1944

|ab|=3^{3+2}(4\cdot2)=3^5\cdot8,\quad3\nmid8\Rightarrow v_3(ab)=5

a+b=-90

90=3^2\cdot10,\quad3\nmid10\Rightarrow v_3(a+b)=2=\min(3,2)

a=p^ru,\quad b=p^sv,\quad r<s,\quad p\nmid u,v

a+b=p^r(u+p^{s-r}v),\quad u+p^{s-r}v\equiv u\not\equiv0\pmod p

v_3(3)=v_3(6)=1

v_3(3+6)=v_3(9)=2>1

-108+108=0

v_3(0)=\texttt{infinity},\quad\texttt{infinity}>k\ (k\in\mathbb N_0)

v_p(a)=r,\quad v_p(b)=s\text{ finite}

p^{\min(r,s)}\mid a+b\Rightarrow v_p(a+b)\ge\min(r,s)

\text{base }4,\quad a=b=2

v_4(2)=0,\quad v_4(4)=1\ne0+0

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S09 · premise_audit · G1, S03, S08 — This does not refute prime-3 certificates; it shows the prime-base premise is essential.
• S08 · premise_audit · G1, S05, S06, S07 — Only unequal r,s guarantee equality by S05; equal r,s cannot be upgraded.
• S07 · premise_audit · G1, S01, S06 — If one input is zero, the product is zero and has valuation infinity; the sum equals the other input, so its valuation equals that input's valuation. If both inputs are zero, the sum is zero and its valuation is infinity, consistently with these conventions.
• S06 · premise_audit · G1, S05 — This refutes unconditional equality while respecting the sum inequality.
• S05 · premise_audit · G1, S04 — a,b and their unit parts may be negative; nondivisibility still applies.
• S04 · premise_audit · G1, S01, S02 — Normalize -90 by absolute value rather than approximate logarithms.
• S03 · premise_audit · G1, S01, S02 — The unit product 8 is not divisible by 3; merely showing 3^5 divides would be insufficient.
• S02 · premise_audit · G1, S01 — Both inputs are nonzero, so their finite valuations 3,2 can be added.
• S01 · premise_audit · G1 — Stopping at 12 would underestimate the valuation; maximality needs the final nondivisibility check.

Error controls and negative controls

Highest power

After |a|=p^k*u, check p does not divide u before assigning k.

Zero sentinel

Zero uses the string infinity only, not 0 or an unmarked NaN.

• Negative integers have no valuation. → reject — Extraction uses absolute value, giving -108 value 3.
• Every sum valuation equals the minimum. → reject — 3+6=9 gives 2>1.
• Composite base 4 preserves the product law. → reject — 2*2=4 gives 1 distinct from 0+0.

Result and limitations

v3(-108)=3,v3(18)=2,v3(-1944)=5,v3(-90)=2; zero uses infinity.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 95c4e2650117d3d500f2fb521401fae53535c51db055a8d4462a820e89c80a8d
• original source SHA-256: 1baa04ecf21c750d6a25a649db9321d708caa5c1eac27fabb94edf46138ffe75
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)