Chevyshev Said It and Ill Say It Again

May 17, 2022 Post a Comment

Bertrand's Postulate

Bertrand'southward postulate, too chosen the Bertrand-Chebyshev theorem or Chebyshev's theorem, states that if , at that place is always at least one prime between and . Equivalently, if , and then there is ever at to the lowest degree one prime such that north<p<2n . The theorize was start fabricated past Bertrand in 1845 (Bertrand 1845; Nagell 1951, p. 67; Havil 2003, p. 25). Information technology was proved in 1850 past Chebyshev (Chebyshev 1854; Havil 2003, p. 25; Derbyshire 2004, p. 124) using non-uncomplicated methods, and is therefore sometimes known as Chebyshev'south theorem. The first simple proof was by Ramanujan, and later improved past a xix-year-former Erdős in 1932.

A short verse about Bertrand's postulate states, "Chebyshev said it, but I'll say it again; There'due south e'er a prime between and ." While unremarkably 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 consequence is that if , then in that location is a number containing a prime divisor in the sequence , , ..., . (The case then corresponds to Bertrand'southward postulate.) This was first proved by Sylvester, independently past 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, 1, 1, 1, 1, ii, 2, 3, 3, 3, iii, 3, 3, 4, ... (OEIS A077463), while the numbers of primes betwixt and are 0, one, 1, 2, one, ii, two, ii, 3, iv, 3, 4, 3, 3, ... (OEIS A060715). For , ii, ..., the values of , where is the adjacent prime function are 2, 3, five, v, 7, 7, 11, eleven, eleven, 11, 13, thirteen, 17, 17, 17, 17, 19, ... (OEIS A007918).

After his proof of Bertrand's postulate, Ramanujan (1919) proved the generalization that , ii, 3, iv, 5, ... if , 11, 17, 29, 41, ... (OEIS A104272), respectively, where is the prime counting office. The numbers are sometimes known as Ramanujan primes. The case for all is Bertrand'due south postulate.

A related trouble is to observe the least value of so that there exists at least one prime between and for sufficiently big (Berndt 1994). The smallest known value is (Lou and Yao 1992).

Explore with Wolfram|Blastoff

References

Aigner, M. and Ziegler, G. M. Proofs from the Book, 2nd ed. New York: Springer-Verlag, 2000. Berndt, B. C. Ramanujan's 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. 50'École Roy. Polytech. 17, 123-140, 1845. Chebyshev, P. "Mémoire sur les nombres premiers." Mém. Acad. Sci. St. Pétersbourg 7, 17-33, (1850) 1854. Reprinted equally §1-7 in Œuvres de P. L. Tschebychef, Tome I. St. Pétersbourg, Russian federation: Commissionaires de fifty'Academie Impériale des Sciences, pp. 51-64, 1899. Derbyshire, J. Prime number Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004. Dickson, L. E. "Bertrand's 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 Conference held at Anna University, Madras, Dec. 21, 1987. (Ed. M. Alladi). New York: Springer-Verlag, pp. ane-20, 1989. Erdős, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934. Hardy, G. H. and Wright, E. Thou. "Bertrand's Postulate and a 'Formula' for Primes." §22.three in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Printing, pp. 343-345 and 373, 1979. Havil, J. Gamma: Exploring Euler'southward Constant. Princeton, NJ: Princeton University Press, 2003. Hoffman, P. The Man Who Loved Simply 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 (II)." Hardy-Ramanujan J. xv, i-33, 1992. Nagell, T. Introduction to Number Theory. New York: Wiley, p. 70, 1951. Ramanujan, S. "A Proof of Bertrand's Postulate." J. Indian Math. Soc. 11, 181-182, 1919. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., pp. 208-209, 2000. Schechter, B. My Encephalon is Open: The Mathematical Journeys of Paul Erdős. New York: Simon and Schuster, 1998. Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. seven-viii, 2000. Sloane, N. J. A. Sequences A007918, A060715, A077463, and A104272 in "The On-Line Encyclopedia of Integer Sequences."

Cite this equally:

Sondow, Jonathan and Weisstein, Eric W. "Bertrand'due south Postulate." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BertrandsPostulate.html