Araştırma Makalesi
BibTex RIS Kaynak Göster

Hales-Jewett Teoremi ile Asalların Sonsuzluğu

Yıl 2025, Cilt: 10 Sayı: 2, 439 - 449, 24.12.2025

Azem Berivan Adıbelli , Sadık Eyidoğan

https://doi.org/10.33484/sinopfbd.1643702

Öz

Bu çalışmada, asal sayıların sonsuzluğunu kanıtlamak için Hales–Jewett Teoremi kullanılarak farklı bir yol izlenmektedir. Bu yaklaşım, asal sayıların dağılımını incelerken, onların belirli kombinatoriyel yapılar içinde nasıl yer aldığını ortaya koyar. Böylece asal sayıların yalnızca klasik aritmetik yöntemlerle değil, aynı zamanda daha geniş yapısal çerçeveler içinde de ele alınabileceği gösterilmiş olur.

Anahtar Kelimeler

Asalların sonsuzluğu Hales-Jewett Teoremi Ramsey teorisi

Etik Beyan

Çalışma, etik kurul izni ve herhangi bir özel izin gerektirmemektedir

Destekleyen Kurum

Bu çalışma kısmen Türkiye Bilimsel ve Teknolojik Araştırma Kurumu tarafından TÜBİTAK-122F027 proje numarasıyla desteklenmiştir.

Teşekkür

Yazarlar, yorumları ve önerileriyle bu makalenin geliştirilmesine ve netleştirilmesine katkıda bulunan hakemlere teşekkür eder. Ayrıca, bu çalışmanın dahil olduğu TÜBİTAK projesinin yürütücüsü Haydar GÖRAL’a desteklerinden dolayı teşekkür eder.

Kaynakça

  • Ramsey, F. P. (1930). On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264-286.
  • Schur, I. (1916). Über die kongruenz xm + ym ≡ zm (mod p). Jahresbericht der Deutschen Mathematiker-Vereinigung 25.
  • van derWaerden, B. L. (1927). Beweis einer baudetschen vermutung. Nieuw Archief voorWiskunde 15, 212-216.
  • Hales, A.W., & Jewett, R. I. (1963). Regularity and positional games. Transactions of the American Mathematical Society 106, 222–229.
  • Näslund, M. (2013). The Hales-Jewett Theorem and its application to further generalizations of m, n, k-games.
  • Erd˝os, P., & Turán, P. (1936). On some sequences of integers. Journal of the London Mathematical Society 11, 261–264. https://doi.org/10.1112/jlms/s1-11.4.261
  • Roth, K. F. (1953). On certain sets of integers. Journal of the London Mathematical Society, 28, 104–109. https://doi.org/10.1112/jlms/s1-28.1.104
  • Szemerédi, E. (1975). On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27, 199–245.
  • Alpoge, L. (2015). Van der Waerden and the primes. American Mathematical Monthly 122, 784–785. https://doi.org/10.4169/amer.math.monthly.122.8.784
  • Granville, A. (2017). Squares in arithmetic progressions and infinitely many primes. American Mathematical Monthly, 124, 951–954. doi.org/10.4169/amer.math.monthly.124.10.951
  • Elsholtz, C. (2020). Fermat’s Last Theorem implies Euclid’s infinitude of primes. American Mathematical Monthly, 128, 250–257. https://doi.org/10.1080/00029890.2021.1856544
  • Göral, H., Özcan, H. B., & Sertba¸s, D. C. (2022). The Green-Tao Theorem and the infinitude of primes in domains. American Mathematical Monthly, 130, 114–125.
  • Gasarch, W. (2023). Fermat’s Last Theorem, Schur’s Theorem (in Ramsey theory), and the infinitude of the primes. Discrete Mathematics, 346. https://doi.org/10.1016/j.disc.2023.113877
  • Meštrovi´c, R. (2023). Euclid’s Theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.-2022). https://doi.org/10.48550/arXiv.1202.3670
  • Göral, H. (2020). p−adic metrics and the infinitude of primes. Mathematics Magazine 93, 19–22.
  • Göral, H., & Özcan, H. B. (2020). Several novel proofs of the infinitude of primes. Mathematics Student 89, 91–95.
  • Gasarch, W. (2024). A webpage of papers that prove primes infinite using Ramsey Theory. Retrieved from https://www.cs.umd.edu/ gasarch/TOPICS/ramseyprimes/ramseyprimes.html
  • Hales, A. W., & Jewett, R. I. (2009). Regularity and positional games. In Classic Papers in Combinatorics (pp. 320-327). Boston, MA: Birkhäuser Boston.
  • Gouvêa, F. Q. (2020). p-adic Numbers: An Introduction. Third edition, Springer.

The Infinitude of the Primes via the Hales-Jewett Theorem

Yıl 2025, Cilt: 10 Sayı: 2, 439 - 449, 24.12.2025

Azem Berivan Adıbelli , Sadık Eyidoğan

https://doi.org/10.33484/sinopfbd.1643702

Öz

We take a different approach to prove the infinitude of the prime numbers by utilizing the Hales–Jewett Theorem. This method allows us to examine the distribution of primes while observing how they are situated within specific combinatorial structures. In this way, we demonstrate that primes can be studied not only through classical arithmetic methods but also within broader structural frameworks.

Anahtar Kelimeler

Hales-Jewett theorem Infinitude of the primes Ramsey Theory

Kaynakça

  • Ramsey, F. P. (1930). On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264-286.
  • Schur, I. (1916). Über die kongruenz xm + ym ≡ zm (mod p). Jahresbericht der Deutschen Mathematiker-Vereinigung 25.
  • van derWaerden, B. L. (1927). Beweis einer baudetschen vermutung. Nieuw Archief voorWiskunde 15, 212-216.
  • Hales, A.W., & Jewett, R. I. (1963). Regularity and positional games. Transactions of the American Mathematical Society 106, 222–229.
  • Näslund, M. (2013). The Hales-Jewett Theorem and its application to further generalizations of m, n, k-games.
  • Erd˝os, P., & Turán, P. (1936). On some sequences of integers. Journal of the London Mathematical Society 11, 261–264. https://doi.org/10.1112/jlms/s1-11.4.261
  • Roth, K. F. (1953). On certain sets of integers. Journal of the London Mathematical Society, 28, 104–109. https://doi.org/10.1112/jlms/s1-28.1.104
  • Szemerédi, E. (1975). On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27, 199–245.
  • Alpoge, L. (2015). Van der Waerden and the primes. American Mathematical Monthly 122, 784–785. https://doi.org/10.4169/amer.math.monthly.122.8.784
  • Granville, A. (2017). Squares in arithmetic progressions and infinitely many primes. American Mathematical Monthly, 124, 951–954. doi.org/10.4169/amer.math.monthly.124.10.951
  • Elsholtz, C. (2020). Fermat’s Last Theorem implies Euclid’s infinitude of primes. American Mathematical Monthly, 128, 250–257. https://doi.org/10.1080/00029890.2021.1856544
  • Göral, H., Özcan, H. B., & Sertba¸s, D. C. (2022). The Green-Tao Theorem and the infinitude of primes in domains. American Mathematical Monthly, 130, 114–125.
  • Gasarch, W. (2023). Fermat’s Last Theorem, Schur’s Theorem (in Ramsey theory), and the infinitude of the primes. Discrete Mathematics, 346. https://doi.org/10.1016/j.disc.2023.113877
  • Meštrovi´c, R. (2023). Euclid’s Theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.-2022). https://doi.org/10.48550/arXiv.1202.3670
  • Göral, H. (2020). p−adic metrics and the infinitude of primes. Mathematics Magazine 93, 19–22.
  • Göral, H., & Özcan, H. B. (2020). Several novel proofs of the infinitude of primes. Mathematics Student 89, 91–95.
  • Gasarch, W. (2024). A webpage of papers that prove primes infinite using Ramsey Theory. Retrieved from https://www.cs.umd.edu/ gasarch/TOPICS/ramseyprimes/ramseyprimes.html
  • Hales, A. W., & Jewett, R. I. (2009). Regularity and positional games. In Classic Papers in Combinatorics (pp. 320-327). Boston, MA: Birkhäuser Boston.
  • Gouvêa, F. Q. (2020). p-adic Numbers: An Introduction. Third edition, Springer.
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Konular Cebir ve Sayı Teorisi, Kombinatorik ve Ayrık Matematik (Fiziksel Kombinatorik Hariç)
Bölüm Araştırma Makalesi
Yazarlar

Azem Berivan Adıbelli 0009-0005-7995-1769

Sadık Eyidoğan 0000-0003-4324-9845

Gönderilme Tarihi 20 Şubat 2025
Kabul Tarihi 8 Eylül 2025
Yayımlanma Tarihi 24 Aralık 2025
Yayımlandığı Sayı Yıl 2025 Cilt: 10 Sayı: 2

Kaynak Göster

APA Adıbelli, A. B., & Eyidoğan, S. (2025). Hales-Jewett Teoremi ile Asalların Sonsuzluğu. Sinop Üniversitesi Fen Bilimleri Dergisi, 10(2), 439-449. https://doi.org/10.33484/sinopfbd.1643702


Sinopfbd' de yayınlanan makaleler CC BY-NC 4.0 ile lisanslanmıştır.