Erdos Said and Ill Say It Again
Bertrand'south Postulate
Bertrand'due south postulate, likewise called the Bertrand-Chebyshev theorem or Chebyshev'south theorem, states that if 
, there is always at to the lowest degree one prime number 
between 
and 
. Equivalently, if 
, then there is always at to the lowest degree one prime 
such that 
. The conjecture was beginning made by Bertrand in 1845 (Bertrand 1845; Nagell 1951, p. 67; Havil 2003, p. 25). It was proved in 1850 by Chebyshev (Chebyshev 1854; Havil 2003, p. 25; Derbyshire 2004, p. 124) using non-uncomplicated methods, and is therefore sometimes known as Chebyshev's theorem. The offset uncomplicated proof was by Ramanujan, and afterwards improved by a 19-year-old Erdős in 1932.
A short verse about Bertrand's postulate states, "Chebyshev said information technology, just I'll say information technology again; At that place's ever a prime between 
and 
." While usually attributed to Erdős or to some other Hungarian mathematician upon Erdős'south youthful re-proof the theorem (Hoffman 1998), the quote is really due to N. J. Fine (Schechter 1998).
An extension of this outcome is that if 
, then there is a number containing a prime divisor 
in the sequence 
, 
, ..., 
. (The example 
and so corresponds to Bertrand'south postulate.) This was starting time proved by Sylvester, independently by Schur, and a simple proof was given by Erdős (1934; Hoffman 1998, p. 37)
The numbers of primes betwixt 
and 
for 
, 2, ... are 0, 0, 0, one, i, 1, ane, ii, ii, three, 3, 3, 3, 3, three, four, ... (OEIS A077463), while the numbers of primes between 
and 
are 0, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, iii, iii, ... (OEIS A060715). For 
, 2, ..., the values of 
, where 
is the side by side prime function are 2, 3, 5, 5, 7, 7, 11, 11, eleven, 11, 13, 13, 17, 17, 17, 17, 19, ... (OEIS A007918).
After his proof of Bertrand's postulate, Ramanujan (1919) proved the generalization that 
, two, 3, 4, 5, ... if 
, xi, 17, 29, 41, ... (OEIS A104272), respectively, where 
is the prime counting role. The numbers are sometimes known as Ramanujan primes. The case 
for all 
is Bertrand's postulate.
A related problem is to find the least value of 
so that in that location exists at least ane prime between 
and 
for sufficiently big 
(Berndt 1994). The smallest known value is 
(Lou and Yao 1992).
See also
Choquet Theory, de Polignac's Conjecture, Landau's Problems, Next Prime number, Prime number Number, Ramanujan Prime
Portions of this entry contributed past Jonathan Sondow (author'due south link)
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References
Aigner, Chiliad. and Ziegler, G. Thousand. Proofs from the Book, 2nd ed. New York: Springer-Verlag, 2000. Berndt, B. C. Ramanujan'south Notebooks, Part IV. New York: Springer-Verlag, p. 135, 1994. Bertrand, J. "Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme." J. l'École Roy. Polytech. 17, 123-140, 1845. Chebyshev, P. "Mémoire sur les nombres premiers." Mém. Acad. Sci. St. Pétersbourg seven, 17-33, (1850) 1854. Reprinted equally §1-7 in Œuvres de P. L. Tschebychef, Tome I. St. Pétersbourg, Russian federation: Commissionaires de l'Academie Impériale des Sciences, pp. 51-64, 1899. Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Trouble in Mathematics. New York: Penguin, 2004. Dickson, L. E. "Bertrand'due south Postulate." History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 435-436, 2005. Erdős, P. "Ramanujan and I." In Proceedings of the International Ramanujan Centenary Briefing held at Anna Academy, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer-Verlag, pp. 1-xx, 1989. Erdős, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934. Hardy, G. H. and Wright, E. M. "Bertrand's Postulate and a 'Formula' for Primes." §22.iii in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 343-345 and 373, 1979. Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton Academy Press, 2003. Hoffman, P. The Man Who Loved But Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998. Lou, S. and Yau, Q. "A Chebyshev's Type of Prime number Theorem in a Short Interval (2)." Hardy-Ramanujan J. xv, one-33, 1992. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 70, 1951. Ramanujan, Southward. "A Proof of Bertrand'due south Postulate." J. Indian Math. Soc. 11, 181-182, 1919. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. South. Aiyar, and B. G. Wilson). Providence, RI: Amer. Math. Soc., pp. 208-209, 2000. Schechter, B. My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: Simon and Schuster, 1998. Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. vii-8, 2000. Sloane, N. J. A. Sequences A007918, A060715, A077463, and A104272 in "The On-Line Encyclopedia of Integer Sequences."
Cite this every bit:
Sondow, Jonathan and Weisstein, Eric Westward. "Bertrand's Postulate." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BertrandsPostulate.html
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