The average gap between primes near a large number is approximately 2.3 times its digit count.
4 events · 1 assessment · 2 decisions
Structured and assessed
First pass. The claim is a popularized corollary of the prime number theorem. Decomposition: one 'requires' edge to the prime number theorem, which the Matcher (ten framings including the negation, the Li(x) form and the p_n ~ n ln n corollary) reported absent from the graph; created it as a mathematical claim tagged mathematics at importance 0.15 / contestation 0 so it stays a deferred stub, seeded at credence 1. The conversion ln N = ln 10 * log10 N and the digit-count bound are derivation steps and were left in prose per §6. Read the Quanta source whole: passage verbatim, byline Erica Klarreich, 2013-05-19; filled in the instance's missing speaker/publication/date and raised its confidence. One web search surfaced the Prime Pages stating the same proposition in ln units; recorded as an affirming instance at confidence 0.7 given the unit difference. Provenance reading recorded; map written immaterial (two textbook restatements, no chain). Importance set to 0.12, contestation 0.02: settled theorem, nothing live turns on it. Canonical form left unchanged: sixteen words, neutral, frame-independent, and the 'near a large number' qualifier is load-bearing so should not be trimmed. No formal statement drafted: the domain skill directs formalization at open claims in the notable range, and Mathlib at the pin does not carry the full prime number theorem, so this settled low-importance claim does not warrant it. Assessed verified, credence 0.99, marginal yield 0.02. No dependents exist, so no notification. Noted for the Curator's attention only in passing: the Matcher observed that the existing RH claim eb2ad3d6-9743-42e1-a264-f6a6bed2bf87 references 'the prime number theorem's average' and might want to link to the new PNT node; that is a cross-claim edge in the Curator's domain and not urgent, so not escalated separately.
Assessed Verified
verdict confidence 0.97 · credence 0.99
The claim is a restatement, in decimal-digit units, of the prime number theorem. Because the number of primes up to x is asymptotically x divided by the natural logarithm of x, the average spacing between consecutive primes near a large number N is asymptotically ln N. The natural logarithm of N equals ln 10, about 2.3026, times the base-ten logarithm of N, and the base-ten logarithm of N is within one of the number of decimal digits of N. So the average gap near N is about 2.3 times its digit count: among 100-digit numbers, for example, the average gap is about 230. The theorem underlying this has been proved since 1896 (Hadamard and de la Vallée Poussin, with elementary proofs by Selberg and Erdős in 1949) and no informed party disputes it. The word "approximately" carries two harmless slacks: the ratio ln N divided by the digit count tends to 2.3026 only as N grows, and the digit count exceeds log10 N by a fraction between 0 and 1. Neither affects the claim as a statement about large numbers. The claim concerns the average only; individual gaps range from 2 (twin primes) to far above the average, which is exactly the contrast the sources that state it go on to draw.
Updated claim instance
Instance 93d3b58c-752c-4e74-b680-d9083091b70c (https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/): was stance=affirms, confidence=0.8; set confidence=0.95, speaker="Erica Klarreich", publication="Quanta Magazine", sourceDate="2013-05-19". Read the source whole. The byline is Erica Klarreich, Contributing Correspondent, Quanta Magazine, dated May 19, 2013; the recorded passage appears verbatim in the article's "The Problem of Pairs" section, stated in the author's own voice as an established fact ("For over a century, mathematicians have understood..."). Metadata was missing and is filled in; confidence raised since the passage is an unambiguous assertion of the claim.
Claim entered the graph