How Erdos Proved a Prime Sits Between n and 2n
Pick any whole number, double it, and look at the stretch in between. There is always a prime hiding in there — between n and 2n, every time, no exceptions, all the way up the number line. That is strange, because primes get rarer the further out you go, and the gaps between them grow without limit, so a doubling window really could have come up empty. It never does. This is Bertrand's postulate, and the reason comes from a completely unexpected place: a counting number from the middle of Pascal's triangle, the central binomial coefficient C(2n, n), squeezed so tightly between two known sizes that it is forced to contain a fresh prime just to be as large as it has to be.
This video walks the whole proof — the elementary one Paul Erdos published in 1932, at nineteen, in his first paper.
This is part of the series: How X Solved Y.
We cover:
• the doubling window: pick n, double to 2n, and the gap (n, 2n] always holds a prime (5 → 7, and (10,20] holds 11, 13, 17, 19)
• why it looks like it should fail: primes thin out (25% → 17% → 10%), the gaps grow without bound
• the brute-force check: every n up to 2,000 has a prime in its window, and the count climbs (the window after 1,000 holds 135 primes)
• the surprising tool: the central binomial coefficient C(2n, n), squeezed by 4ⁿ/(2n+1) ≤ C(2n, n) ≤ 4ⁿ
• what that number is made of: C(30, 15) = 155,117,520 = 2⁴·3²·5·17·19·23·29, and the three facts about its prime factors
• the collision: if the window were empty, C(2n, n) could not be as large as it provably is — so a prime is forced
• the history: Bertrand (1845), Chebyshev's first proof (1852), and Erdos's elementary proof (1932)
CHAPTERS
0:00 The Doubling Window
1:15 Why It Should Fail
2:33 Brute Force First
4:03 The Surprising Tool
5:33 What the Number Is Made Of
7:13 The Collision
8:54 The Three Who Settled It
10:12 The Count Keeps Climbing
11:40 Coda
Every number on screen is computed, not estimated: the smallest prime in each window (n, 2n], the prime counts π(2n) − π(n), the central binomial coefficients C(2n, n) and their bounds, the full factorization of C(30, 15), and the Bertrand prime chain 2, 3, 5, 7, 13, 23, 43, 83, … that covers the small cases. Erdos summed the whole story up in a rhyme: "Chebyshev said it, and I say it again, there is always a prime between n and 2n." One open question sits right next door: whether there is always a prime between consecutive squares, n² and (n+1)², is still unproven today.
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Music by Vincent Rubinetti
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https://vincerubinetti.bandcamp.com/a...