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Brocard's Conjecture

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Brocard's conjecture (Brocard 1904) states that

pi(p_(n+1)^2)-pi(p_n^2)>=4

for n>=2, where pi(n) is the prime counting function and p_n is the nth prime.

The folowing table lists successive primes betwen succesive values of p_n^2.

np_np_(n+1)primes between p_n^2 and p_(n+1)^2number of primes
1235, 72
23511, 13, 17, 19, 235
35729, 31, 37, 41, 43, 476
471153, 59, 61, 67, 71, 73, 79, 83, 89, 98, 101, 103, 107, 109, 11315
51113127, 131, 137, 139, 149, 151, 157, 163, 1679
BrocardsConjecture

For n=1, 2, ..., the first few values are 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, ... (OEIS A050216), illustrated above.

The high water marks for these values are 2, 5, 6, 15, 22, 27, 47, 57, 80, ... (OEIS A380135), which occur for n=1, 2, 3, 4, 6, 8, 9, 11, 15, 18, 21, (OEIS A380136).

BrocardsConjectureScatterDiagram

When plotted as a scatter diagram, a sequence of rays radiating from the origin becomes evident.

The conjecture is true for 2<=n<=4×10^5 (E. Weisstein, Jan. 12, 2025) but remains unproven.

See also

Andrica's Conjecture, Cramér Conjecture, Legendre's Conjecture

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References

Brocard, H. Response to Problem 2181 in L'intermédiaire des math. 11, 149, 1904.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 436, 2005.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 248, 1989.Shannon, A. G. and Leyendekkers, J. V. "On Legendre's Conjecture." Notes Number Th. Disc. Math. 23, 117-125, 2017.Sloane, N. J. A. Sequences A050216, A380135, and A380136 in "The On-Line Encyclopedia of Integer Sequences."

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Brocard's Conjecture

Cite this as:

Weisstein, Eric W. "Brocard's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardsConjecture.html

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