# On the greatest prime factor of ${n}^{2}+1$

Jean-Marc Deshouillers; Henryk Iwaniec

Annales de l'institut Fourier (1982)

- Volume: 32, Issue: 4, page 1-11
- ISSN: 0373-0956

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topDeshouillers, Jean-Marc, and Iwaniec, Henryk. "On the greatest prime factor of $n^2+1$." Annales de l'institut Fourier 32.4 (1982): 1-11. <http://eudml.org/doc/74560>.

@article{Deshouillers1982,

abstract = {There exist infinitely many integers $n$ such that the greatest prime factor of $n^2+1$ is at least $n^\{6/5\}$. The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.},

author = {Deshouillers, Jean-Marc, Iwaniec, Henryk},

journal = {Annales de l'institut Fourier},

keywords = {greatest prime factor; combination of Hooley's method; upper bound for Kloosterman sums},

language = {eng},

number = {4},

pages = {1-11},

publisher = {Association des Annales de l'Institut Fourier},

title = {On the greatest prime factor of $n^2+1$},

url = {http://eudml.org/doc/74560},

volume = {32},

year = {1982},

}

TY - JOUR

AU - Deshouillers, Jean-Marc

AU - Iwaniec, Henryk

TI - On the greatest prime factor of $n^2+1$

JO - Annales de l'institut Fourier

PY - 1982

PB - Association des Annales de l'Institut Fourier

VL - 32

IS - 4

SP - 1

EP - 11

AB - There exist infinitely many integers $n$ such that the greatest prime factor of $n^2+1$ is at least $n^{6/5}$. The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.

LA - eng

KW - greatest prime factor; combination of Hooley's method; upper bound for Kloosterman sums

UR - http://eudml.org/doc/74560

ER -

## References

top- [1] J.-M. DESHOUILLERS and H. IWANIEC, Kloosterman sums and Fourier coefficients of cusp forms, Inv. Math. (to appear). Zbl0502.10021
- [2] C. HOOLEY, On the greatest prime factor of a quadratic polynomial, Acta Math., 117 (1967), 281-299. Zbl0146.05704MR34 #4225
- [3] C. HOOLEY, Applications of sieve methods to the theory of numbers, Cambridge Univ. Press, London, 1976. Zbl0327.10044
- [4] H. IWANIEC, Rosser's sieve, Acta Arith., 36 (1980), 171-202. Zbl0435.10029
- [5] H.J.S. SMITH, Report on the theory of numbers, Collected Mathematical Papers, vol. I, reprinted, Chelsea, 1965.

## Citations in EuDML Documents

top- J. Pomykała, On the greatest prime divisor of quadratic sequences
- Jacek Pomykała, Cubic norms represented by quadratic sequences
- Cécile Dartyge, Le plus grand facteur premier de n² + 1 où est presque premier
- C. Dartyge, Propriétés multiplicatives des valeurs de certains polynômes en deux variables
- Adolf Hildebrand, Gerald Tenenbaum, Integers without large prime factors
- János Pintz, Landau’s problems on primes

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