The average gap between primes near a large number is approximately 2.3 times its digit count.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim turns on a single dependency, the prime number theorem, which is settled mathematics. The derivation from it is short and is checked here directly rather than taken from any source. If pi(x) ~ x/ln x, then between x and x + h (for h small relative to x but large relative to ln x) there are about h/ln x primes, so the mean gap is ln x. For N with d decimal digits, 10^(d-1) <= N < 10^d, so ln N lies between (d-1) ln 10 and d ln 10, i.e. between about 2.30(d-1) and 2.30d. For d = 100 this gives a mean gap between about 228 and 230, matching the illustrative figure of about 230 that Quanta gives.
Both recorded instances affirm. Erica Klarreich's Quanta Magazine article of 19 May 2013 (www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/) states it in the digit-count form verbatim as recorded, as established background before describing Zhang's bounded-gaps result. The Prime Pages note on prime gaps (t5k.org/notes/gaps.html) states the same proposition in natural-logarithm units, that the average gap near n is about log n. No source denying the proposition exists, and none could, since it is a theorem's corollary.
Two readings of "approximately" were considered as possible ways the claim could fail: an exact-constant reading (the ratio is 2.3026..., not 2.3) and a small-N reading (for numbers with few digits the ratio is noticeably off, e.g. the mean gap among two-digit numbers is about 4, not 4.6). The claim's own words, "near a large number", exclude both. What would change the verdict is nothing short of a refutation of the prime number theorem, which is not a live possibility.
Decomposition
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The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →The number of primes up to x is asymptotically x divided by the natural logarithm of x. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
Among large numbers, the expected gap between prime numbers is approximately 2.3 times the number of digits
Describing how primes taper off on average.
Asserted without evidence of the source's own. The article states the average gap as established background, without citing the prime number theorem by name or giving a derivation, and illustrates it with the example that among 100-digit numbers the expected gap is about 230. The statement is an accurate popularization of the theorem.
we expect much more since the "average gap" is about log(n)
Explaining prime gaps for a general mathematical audience; states the average gap near n as about log n (the natural logarithm), which equals about 2.3 times the number of decimal digits of n. Same proposition in different units.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.