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On shifted primes with large prime factors and their products

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Abstract

In this short note, we give partial answers to two questions on shifted primes with large prime factors, posed by Luca et al. (Bull Belg Math Soc Simon Stevin 22:39–47, 2015) and by Chen and Chen (Acta Math Sin (Engl Ser) 33(3):377–382, 2017), respectively.

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References

  1. Baker, R.C., Harman, G.: The Brun–Titchmarsh theorem on average. In: Proceedings Conference in Honor of Heini Halberstam (Allerton Park, IL, 1995), Progress in Mathematics, vol. 138, pp. 39–103. Birkhäuser, Boston (1996)

  2. Banks, W., Shparlinski, Igor E.: On values taken by the largest prime factor of shifted primes. J. Aust. Math. Soc. 82, 133–147 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Billerey, N., Menares, R.: On the modularity of reducible mod \(\ell \) Galois representations. Math. Res. Lett. 23(1), 15–41 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, F.-J., Chen, Y.-G.: On the largest prime factor of shifted primes. Acta Math. Sin. (Engl. Ser.) 33(3), 377–382 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng, B., Wu, J.: On the density of shifted primes with large prime factors. Sci. China Math. 12(1), 83–94 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fouvry, É.: Théorème de Brun–Titichmarsh; application au théorème de Fermat. Invent. Math. 79, 383–407 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Granville, A.: Smooth numbers: computational number theory and beyond. In: Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, vol. 44, pp. 267–323. Cambridge University Press, Cambridge (2008)

  8. Liu, J.-Y., Wu, J., Xi, P.: Primes in arithmetic progressions with friable indices. Preprint (2017)

  9. Luca, F., Menares, R., Pizarro-Madariaga, A.: On shifted primes with large prime factors and their products. Bull. Belg. Math. Soc. Simon Stevin 22, 39–47 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Mikawa, H.: On primes in arithmetic progressions. Tsukuba J. Math. 25, 121–153 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pomerance, C.: Popular values of Euler’s function. Mathematika 27, 84–89 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, Z.-W.: Autour des plus grands facteurs premiers d’entiers consécutifs voisins d’un entier criblé. Q. J. Math. 69(3), 995–1013 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Y.: Bounded gaps between primes. Ann. Math. (2) 179, 1121–1174 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported in part by Scientific Research Innovation Team Project Affiliated to Yangtze Normal University (No. 2016XJD01).

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Correspondence to Jie Wu.

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Wu, J. On shifted primes with large prime factors and their products. Arch. Math. 112, 387–393 (2019). https://doi.org/10.1007/s00013-018-1272-z

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  • DOI: https://doi.org/10.1007/s00013-018-1272-z

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