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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/122/01/0001-0013

    • Keywords

       

      Additive problems; applications of Bombieri–Vinogradov theorem.

    • Abstract

       

      In 1935, Erdös proved that all natural numbers can be written as a sum of a square of a prime and a square-free number. In 1939, Pillai derived an asymptotic formula for the number of such representations. The mathematical review of Pillai’s paper stated that the proof of the above result contained inaccuracies, thus casting a doubt on the correctness of the paper. In this paper, we re-examine Pillai’s paper and show that his argument was essentially correct. Afterwards, we improve the error term in Pillai’s theorem using the Bombieri–Vinogradov theorem.

    • Author Affiliations

       

      M Ram Murty1 R Thangadurai2

      1. Department ofMathematics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
      2. Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India

    • Dates

       
      Published
      February 2012
      Manuscript received
      9 January 2011
      Manuscript revised
      20 April 2011

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