Are Quadratic Residues Randomly Distributed?

  • Steve Wright
Part of the Lecture Notes in Mathematics book series (LNM, volume 2171)


The purpose of this chapter is to provide evidence that the answer to the question in the title is yes. By examining tables of residues and non-residues of certain primes in Sect. 10.1, we observe that residues can occur in very irregular patterns. In Sect. 10.2, we will show how to view sums of the values of Legendre symbols χ p as random variables and then we will employ the Central Limit Theorem from probability theory to determine a condition under which, at least when p is sufficiently large, the values of χ p can be interpreted to behave randomly and independently. In Sect. 10.3, a very interesting result of Davenport and Erdos on the the distribution of residues will then be employed to verify that the condition from Sect. 10.2 that detects random behavior of residues and non-residues does indeed hold. Interestingly enough, the Weil-sum estimates from Theorem  9.1, which were so useful in our work in Chap.  9, will also be very useful in our proof of Davenport and Erdos’ result.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Steve Wright
    • 1
  1. 1.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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