Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 39-61.

TWO DIVISORS OF (n2+1)/2 SUMMING UP TO δn + δ ± 2, δ EVEN

Sanda Bujačić Babić

Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia
e-mail: sbujacic@math.uniri.hr

Abstract.   We prove there exist infinitely many odd integers n for which there exists a pair of positive divisors d1, d2 of (n2+1)/2 such that

d1 + d2 = δn + ε   for   ε = δ + 2,

where δ is an even positive integer. Furthermore, we deal with the same problem where ε = δ - 2 and δ ≡ 4, 6 (mod 8). Using different approaches and methods we obtain similar but conditional results since the proofs rely on Schinzel’s Hypothesis H.

2010 Mathematics Subject Classification.   11D09, 11A55.

Key words and phrases.   Sum of divisors, continued fractions, Pell equation, Legendre symbol.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/yk3jwhrjd9

References:

  1. M. Ayad, Critical points, critical values of a prime polynomial, Complex Var. Elliptic Equ. 51 (2006), 143-160.
    MathSciNet     CrossRef

  2. M. Ayad and F. Luca, Two divisors of (n2+1)/2 summing up to n + 1, J. Théor. Nombres Bordeaux 19 (2007), 561-566.
    MathSciNet

  3. S. Bujačić, Two divisors of (n2+1)/2 summing up to δn+ε, for δ and ε even, Miskolc Math. Notes 15 (2014), 333-344.
    MathSciNet

  4. A. Dujella and F. Luca, On the sum of two divisors of (n2+1)/2, Period. Math. Hungar. 65 (2012), 83-96.
    MathSciNet     CrossRef

  5. A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185-208, Corrigendum, 5 (1959), 259.
    MathSciNet     CrossRef

Rad HAZU Home Page