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On the sum of the square of a prime and a square-free number

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)16-24
Number of pages9
JournalLMS Journal of Computation and Mathematics
Issue number1
Early online date29 Jan 2016
DateAccepted/In press - 3 Nov 2015
DateE-pub ahead of print - 29 Jan 2016
DatePublished (current) - 1 Feb 2016


We prove that every integer n⩾10 such that n≢1 mod 4 can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author’s numerical computation regarding the generalised Riemann hypothesis to extend the explicit bounds of Ramaré–Rumely.

    Research areas

  • 11N13, 11P32 (primary)

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