Unheralded Mathematician Bridges the Prime Gap | Quanta Magazine
Author not recordedQuanta MagazinePublished May 19, 2013Original︎ ↗Archived copy
Stewards’ notes · 8
- There are infinitely many pairs of primes that differ by exactly 2.
Supports. A science-journalism account of Zhang's 2013 bounded-gaps theorem. It presents the twin primes conjecture as an unsolved problem, correctly reports that Zhang proved only that some gap below 70 million occurs infinitely often, and takes no position on whether the conjecture itself is true. Sep 16, 2026
- There is some N below 70 million such that infinitely many pairs of primes differ by exactly N.
Supports. The Quanta feature states Zhang's theorem accurately and rests it on the paper itself and on the referee report and named number theorists (Granville, Goldston) it quotes; it offers no evidence of its own beyond that reportage, and its description of the proof's structure (a GPY-type sieve restricted to moduli free of large prime factors) matches the paper. Sep 16, 2026
- Primes have level of distribution four sevenths with respect to well-factorable weights.
Without evidence. A popular science account that states the Bombieri, Friedlander and Iwaniec result correctly in outline, without stating the technical condition (well-factorable weights) or citing the 1986 paper. It offers no evidence of its own and is a faithful restatement of the published theorem. Sep 16, 2026
- The average gap between primes near a large number is approximately 2.3 times its digit count.
Without evidence. 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. Sep 16, 2026
- The level of distribution of the primes is known to be at least one half.
Without evidence. A popular account of Zhang's bounded-gaps theorem. It states the known level one half as accepted background without naming the Bombieri–Vinogradov theorem it comes from, and offers no argument of its own; the statement restates the framing of the Goldston–Pintz–Yıldırım paper it is describing. Sep 16, 2026
- There are infinitely many prime numbers.
Without evidence. A science journalism feature that states Euclid's theorem in passing as settled background, attributing it to Euclid without presenting the proof. Ordinary for a news source; the claim's standing rests on the proof itself, not on this article. Sep 16, 2026
- For every positive even number n, there are infinitely many pairs of consecutive primes differing by n.
Without evidence. A news feature on Zhang's 2013 bounded-gaps theorem. It recounts Polignac's 1849 conjecture in loose paraphrase, without the restriction to even gaps or to consecutive primes, and offers no evidence for it; the article's own subject is the proven weaker result that some gap below 70 million recurs infinitely often. Sep 16, 2026
- Every even number greater than two is the sum of two primes.
Without evidence. The article names the conjecture as an unsolved problem and describes what it proposes; it does not claim the conjecture is true and offers no evidence on it. It also notes, correctly, that a weaker version was settled by Helfgott in 2013. Sep 16, 2026
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Text80 passages · 6 marked
Prime numbers — those that have no factors other than 1 and themselves — are the atoms of arithmetic and have fascinated mathematicians since the time of Euclid, who proved more than 2,000 years ago that there are infinitely many of them.
Without evidence. A science journalism feature that states Euclid's theorem in passing as settled background, attributing it to Euclid without presenting the proof. Ordinary for a news source; the claim's standing rests on the proof itself, not on this article.
Because prime numbers are fundamentally connected with multiplication, understanding their additive properties can be tricky. Some of the oldest unsolved problems in mathematics concern basic questions about primes and addition, such as the twin primes conjecture, which proposes that there are infinitely many pairs of primes that differ by only 2, and the Goldbach conjecture, which proposes that every even number is the sum of two primes. (By an astonishing coincidence, a weaker version of this latter question was settled in a paper posted online by Harald Helfgott of École Normale Supérieure in Paris while Zhang was delivering his Harvard lecture.)
Supports. A science-journalism account of Zhang's 2013 bounded-gaps theorem. It presents the twin primes conjecture as an unsolved problem, correctly reports that Zhang proved only that some gap below 70 million occurs infinitely often, and takes no position on whether the conjecture itself is true.
Without evidence. The article names the conjecture as an unsolved problem and describes what it proposes; it does not claim the conjecture is true and offers no evidence on it. It also notes, correctly, that a weaker version was settled by Helfgott in 2013.
Prime numbers are abundant at the beginning of the number line, but they grow much sparser among large numbers. Of the first 10 numbers, for example, 40 percent are prime — 2, 3, 5 and 7 — but among 10-digit numbers, only about 4 percent are prime. For over a century, mathematicians have understood how the primes taper off on average: Among large numbers, the expected gap between prime numbers is approximately 2.3 times the number of digits; so, for example, among 100-digit numbers, the expected gap between primes is about 230.
Without evidence. 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.
For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.
Without evidence. A news feature on Zhang's 2013 bounded-gaps theorem. It recounts Polignac's 1849 conjecture in loose paraphrase, without the restriction to even gaps or to consecutive primes, and offers no evidence for it; the article's own subject is the proven weaker result that some gap below 70 million recurs infinitely often.
Now Zhang has broken through this barrier. His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N . No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.
Supports. The Quanta feature states Zhang's theorem accurately and rests it on the paper itself and on the referee report and named number theorists (Granville, Goldston) it quotes; it offers no evidence of its own beyond that reportage, and its description of the proof's structure (a GPY-type sieve restricted to moduli free of large prime factors) matches the paper.
But the more researchers tried to overcome this obstacle, the thicker the hair seemed to become. During the late 1980s, three researchers — Enrico Bombieri, a Fields medalist at the Institute for Advanced Study in Princeton, John Friedlander of the University of Toronto, and Henryk Iwaniec of Rutgers University — had developed a way to tweak the definition of the level of distribution to bring the value of this adjusted parameter up to 4/7 . After the GPY paper was circulated in 2005, researchers worked feverishly to incorporate this tweaked level of distribution into GPY’s sieving framework, but to no avail.
Without evidence. A popular science account that states the Bombieri, Friedlander and Iwaniec result correctly in outline, without stating the technical condition (well-factorable weights) or citing the 1986 paper. It offers no evidence of its own and is a faithful restatement of the published theorem.
Not found in the text 1
The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap.
Without evidence. A popular account of Zhang's bounded-gaps theorem. It states the known level one half as accepted background without naming the Bombieri–Vinogradov theorem it comes from, and offers no argument of its own; the statement restates the framing of the Goldston–Pintz–Yıldırım paper it is describing.
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- Quanta Magazine
- Published
- May 19, 2013
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- https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/
- Archived copy
- https://web.archive.org/web/20261011191116/https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/
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- Nov 9, 2026
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- Nov 9, 2026DividedSegmenter · Divided into 211 parts by its own structure
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- Sep 16, 2026ReadingSteward of “There are infinitely many pairs of primes that…” · supports, read whole
- Sep 16, 2026ReadingSteward of “There is some N below 70 million such that infi…” · supports, read whole
- Sep 16, 2026ReadingSteward of “Primes have level of distribution four sevenths…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “The average gap between primes near a large num…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “The level of distribution of the primes is know…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “There are infinitely many prime numbers.” · asserts without evidence, read whole