# 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/ln *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/ln *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
\( \le \sqrt x \), and that there are more prime factors
\( \le \sqrt x \) to choose from when *x* is larger.

## Keywords

Number Series Residue Class Successive Prime White Ball Infinite Product## Preview

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## Bibliography

- 1.Daniel Shanks and John W. Wrench Jr., “Brun’s Constant,”
*Math. Comp*.**28**(1974) pp. 293–299.MathSciNetzbMATHGoogle 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 für Mat*.**21**(1983) pp. 45–74.MathSciNetzbMATHCrossRefGoogle Scholar - 5.G. H. Hardy and E. M. Wright,
*An Introduction to the Theory of Numbers*, Fifth edition, Oxford, 1979, p. 351.zbMATHGoogle Scholar - 6.Richard P. Brent, “The Distribution of Small Gaps Between Successive Primes,”
*Math. Comp*.**28**(1974) pp. 315–324.MathSciNetzbMATHCrossRefGoogle Scholar - 7.Ian Richards, “On the Incompatibility of Two Conjectures Concerning Primes,”
*Bull. Amer. Math. Soc*.**80**(1974) pp. 419–438.MathSciNetzbMATHCrossRefGoogle 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 Tidskr. för Informationsbehandling (BIT)***10**(1970) pp. 333–342.MathSciNetzbMATHGoogle Scholar - 10.J. E. Littlewood, “Sur la Distribution des Nombres Premiers,”
*Comptes Rendus***158**(1914) pp. 1869–1872.zbMATHGoogle Scholar - 11.Carter Bays and Richard H. Hudson, “On the Fluctuations of Littlewood for Primes of the Form 4n ± 1,”
*Math. Comp*.**32**(1978) pp. 281–286.MathSciNetzbMATHGoogle Scholar - 12.William Feller,
*An Introduction to Probability Theory and its Applications*, vol. I, Second edition, Wiley, New York, 1957, pp. 73–87.zbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar - 16.Richard P. Brent, “The First Occurrence of Large Gaps Between Successive Primes,”
*Math. Comp*.**27**(1973) pp. 959–963.MathSciNetzbMATHCrossRefGoogle Scholar - 17.Richard P. Brent, “The First Occurrence of Certain Large Prime Gaps,”
*Math. Comp*.**35**(1980) pp. 1435–1436.MathSciNetzbMATHCrossRefGoogle Scholar - 18.Harald Cramér, “On the Order of Magnitude of the Difference Between Consecutive Prime Numbers,”
*Acta Arith*.**2**(1936) pp. 23–46.Google Scholar