# Subtleties in the Distribution of Primes

## Abstract

There are only very few proved results concerning the distribution of primes in short intervals. The prime number theorem tells us that the *average density* of primes around *x* is approximately 1/In *x*. This means that if we consider an interval of length Δ*x* about *x* and choose any integer *t* in this interval, then the probability of *t* being a prime will approach 1/ In *x* as *x* → ∞, if Δ*x* is small compared to *x*. This implies that the primes tend to thin out as *x* grows larger; an implication that becomes obvious when considering that the condition for a randomly picked integer *x* to be composite is that it has some prime factor \( \leqslant \sqrt x\) and that there are more prime factors \( \leqslant \sqrt x\) to choose from when *x* is larger.

## Keywords

Number Series Residue Class White Ball Infinite Product Consecutive Integer## Bibliography

- 1.Daniel Shanks and John W. Wrench, Jr., “Brun’s Constant,”
*Math. Comp.***28**(1974) pp. 293–299.MathSciNetGoogle Scholar - 1′.B. K. Parady, Joel F. Smith, and Sergio E. Zarantonello, “Largest Known Twin Primes,”
*Math. Comp.***55**(1990) pp. 381–382.MathSciNetMATHCrossRefGoogle Scholar - 2.G. H. Hardy and J. E. Littlewood, “Some Problems of ‘Partitio Numerorum’ III: On the Expression of a Number as a Sum of Primes,”
*Acta Math.***44**(1922) pp. 1–70 = G. H. Hardy,*Coll. Papers*, vol. 1, pp. 561–630.MathSciNetCrossRefGoogle Scholar - 3.F. J. Gruenberger and G. Armerding,
*Statistics on the First Six Million Prime Numbers*, Reviewed in*Math. Comp.***19**(1965) pp. 503–505.Google Scholar - 4.Hans Riesel and Robert. C. Vaughan, “On Sums of Primes,”
*Arkiv fr Mat.***21**(1983) pp. 45–74.MathSciNetMATHCrossRefGoogle Scholar - 5.G. H. Hardy and E. M. Wright,
*An Introduction to the Theory of Numbers*, Fifth edition, Oxford, 1979, p. 351.MATHGoogle Scholar - 6.Richard P. Brent, “The Distribution of Small Gaps Between Successive Primes,”
*Math. Comp.***28**(1974) pp. 315–324.MathSciNetMATHCrossRefGoogle Scholar - 7.Ian Richards, “On the Incompatibility of Two Conjectures Concerning Primes,”
*Bull. Am. Math. Soc.***80**(1974) pp. 419–438.MathSciNetMATHCrossRefGoogle Scholar - 8.Thomas Vehka, “Explicit Construction of an Admissible Set for the Conjecture that Sometimes π(
*x*+*y*) > π(*y*) + π(*x*)”*Notices Am. Math. Soc.***26**(1979) p. A-453.Google Scholar - 9.Hans Riesel, “Primes Forming Arithmetic Series and Clusters of Large Primes,”
*Nordisk Tidskrift för Informationsbehandling*(*BIT*)**10**(1970) pp. 333–342.MathSciNetMATHGoogle Scholar - 10.J. E. Littlewood, “Sur la Distribution des Nombres Premiers,”
*Comptes Rendus***158**(1914) pp. 1869–1872.MATHGoogle Scholar - 11.Carter Bays and Richard H. Hudson, “On the Fluctuations of Littlewood for Primes of the Form 4
*n*± 1,”*Math. Comp.***32**(1978) pp. 281–286.MathSciNetMATHGoogle Scholar - 12.William Feller,
*An Introduction to Probability Theory and its Applications*, vol. I, Second edition, Wiley, New York, 1957, pp. 73–87.MATHGoogle Scholar - 13.Carter Bays and Richard H. Hudson, “Details of the First Region of Integers
*x*with π_{3,2}*x*< π_{3,1}*x*”*Math. Comp.***32**(1978) pp. 571–576.MathSciNetMATHGoogle Scholar - 14.Daniel Shanks, “On Maximal Gaps Between Successive Primes,”
*Math. Comp.***18**(1964) pp. 646–651.MathSciNetCrossRefGoogle Scholar - 15.L. J. Lander and T. R. Parkin, “On the First Appearance of Prime Differences,”
*Math. Comp.***21**(1967) pp. 483–488.MathSciNetMATHCrossRefGoogle Scholar - 16.Richard P. Brent, “The First Occurrence of Large Gaps Between Successive Primes,”
*Math. Comp.***27**(1973) pp. 959–963.MathSciNetMATHCrossRefGoogle Scholar - 17.Richard P. Brent, “The First Occurrence of Certain Large Prime Gaps,”
*Math. Comp.***35**(1980) pp. 1435–1436.MathSciNetMATHCrossRefGoogle Scholar - 18.Jeff Young and Aaron Potier, “First Occurrence Prime Gaps,
*Math. Comp.***52**(1989) pp. 221–224.MathSciNetMATHCrossRefGoogle Scholar - 19.Harald Cramér, “On the Order of Magnitude of the Difference Between Consecutive Prime Numbers,”
*Acta Arith.***2**(1936) pp. 23–46.Google Scholar