Giả thuyết Grimm

Trong toán học, cụ thể hơn là trong lý thuyết số, giả thuyết Grimm (đặt tên theo Carl Albert Grimm, 1 tháng 4 năm 1926 – 2 tháng 1 năm 2018) phát biểu rằng mỗi phần tử của tập các hợp số liên tiếp có thể được gán một số nguyên tố p phân biệt là ước của phần tử đó. Bài toán được lần đầu xuất bản trong American Mathematical Monthly, 76(1969) 1126-1128.

Phát biểu

Nếu n + 1, n + 2, …, n + k đều là các hợp số, thì có k số nguyên tố phân biệt pi sao cho pi là ước của n + i với 1 ≤ i ≤ k.

Phiên bản yếu hơn

Phiên bản yếu hơn tuy chưa được chứng minh phát biểu như sau: Nếu không có số nguyên tố trong đoạn , thì có ít nhất k ước nguyên tố phân biệt.

Xem thêm

Tham khảo

  • Erdös, P.; Selfridge, J. L. (1971). “Some problems on the prime factors of consecutive integers II” (PDF). Proceedings of the Washington State University Conference on Number Theory: 13–21.
  • Grimm, C. A. (1969). “A conjecture on consecutive composite numbers”. The American Mathematical Monthly. 76 (10): 1126–1128. doi:10.2307/2317188. JSTOR 2317188.
  • Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 3rd ed., Springer Science+Business Media, pp. 133–134, 2004. ISBN 0-387-20860-7
  • Laishram, Shanta; Murty, M. Ram (2012). “Grimm's conjecture and smooth numbers”. The Michigan Mathematical Journal. 61 (1): 151–160. arXiv:1306.0765. doi:10.1307/mmj/1331222852.
  • Laishram, Shanta; Shorey, T. N. (2006). “Grimm's conjecture on consecutive integers”. International Journal of Number Theory. 2 (2): 207–211. doi:10.1142/S1793042106000498.
  • Ramachandra, K. T.; Shorey, T. N.; Tijdeman, R. (1975). “On Grimm's problem relating to factorisation of a block of consecutive integers”. Journal für die reine und angewandte Mathematik. 273: 109–124. doi:10.1515/crll.1975.273.109.
  • Ramachandra, K. T.; Shorey, T. N.; Tijdeman, R. (1976). “On Grimm's problem relating to factorisation of a block of consecutive integers. II”. Journal für die reine und angewandte Mathematik. 288: 192–201. doi:10.1515/crll.1976.288.192.
  • Sukthankar, Neela S. (1973). “On Grimm's conjecture in algebraic number fields”. Indagationes Mathematicae (Proceedings). 76 (5): 475–484. doi:10.1016/1385-7258(73)90073-5.
  • Sukthankar, Neela S. (1975). “On Grimm's conjecture in algebraic number fields. II”. Indagationes Mathematicae (Proceedings). 78 (1): 13–25. doi:10.1016/1385-7258(75)90009-8.
  • Sukthankar, Neela S. (1977). “On Grimm's conjecture in algebraic number fields-III”. Indagationes Mathematicae (Proceedings). 80 (4): 342–348. doi:10.1016/1385-7258(77)90030-0.
  • Weisstein, Eric W., "Grimm's Conjecture", MathWorld.

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