ON THE NUMBER OF DIVISORS OF $n^{2}-1$

@article{Dudek2015ONTN,
  title={ON THE NUMBER OF DIVISORS OF \$n^\{2\}-1\$},
  author={Adrian W. Dudek},
  journal={Bulletin of the Australian Mathematical Society},
  year={2015},
  volume={93},
  pages={194 - 198}
}
  • Adrian W. Dudek
  • Published 30 July 2015
  • Mathematics
  • Bulletin of the Australian Mathematical Society
We prove an asymptotic formula for the sum $\sum _{n\leq N}d(n^{2}-1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum _{d\leq N}g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_{d}$ to the equation $x^{2}\equiv 1~(\text{mod}~d)$. 
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8 Citations

8 Citations

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Citation Type

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References

SHOWING 1-10 OF 16 REFERENCES
On Diophantine quintuples and $$D(-1)$$D(-1)-quadruples
In this paper the known upper bound $$10^{96}$$1096 for the number of Diophantine quintuples is reduced to $$6.8\cdot 10^{32}$$6.8·1032. The key ingredient for the improvement is that certain…
Bounds on the number of Diophantine quintuples
Multiplicative Number Theory
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The…
There are only finitely many Diophantine quintuples
A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with…
On the number of positive integers $\leq x$ and free of prime factors $>y$
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects…
The greatest prime factor of the integers in a short interval (III)
LetP(x) denote the greatest prime factor of IIz<n≤x+x1/2n. In this paper, we prove thatP(x)>x0.723 holds true for a sufficiently largex.
Van der Corput's method of exponential sums
1. Introduction 2. The simplest Van Der Corput estimates 3. The method of exponent pairs 4. Application of exponent pairs 5. Computing optimal exponent pairs 6. Two dimensional exponential sums 7.…
On the number of divisors of quadratic polynomials
A switch for a suspended railway vehicle having two elastic wheels, comprises four rails forming three paths one of which branches off into two other paths. A gap is bounded between the four rails.…
Some Asymptotic Formulae in the Theory of Numbers
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