{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:59:50Z","timestamp":1760057990896,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T00:00:00Z","timestamp":1741219200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>One of the themes of this paper is recent results on large gaps between primes. The first of these results was achieved in the paper by Ford, Green, Konyagin and Tao. It was later improved in the joint paper of these four authors with Maynard. One of the main ingredients of these results is old methods from Erd\u0151s and Rankin. Other ingredients are important breakthrough results from Goldston, Pintz and Yildirim, and their extension by Maynard on small gaps between primes. All these previous results are discussed in brief. The results on the appearance of k-th powers of primes contained in those large gaps obtained by the author in joint work with Maier are based on a combination of the results just described with the matrix method of Maier.<\/jats:p>","DOI":"10.3390\/axioms14030198","type":"journal-article","created":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T11:11:27Z","timestamp":1741259487000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Recent Results on Large Gaps Between Primes"],"prefix":"10.3390","volume":"14","author":[{"given":"Michael Th.","family":"Rassias","sequence":"first","affiliation":[{"name":"Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,6]]},"reference":[{"unstructured":"Westzynthius, E. (1931). \u00dcber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Commentationes Physico-Mathematicae, Societas Scientiarum Fennica.","key":"ref_1"},{"key":"ref_2","first-page":"1","article-title":"\u00dcber die Differenzen Zwischen den Zahlen, die zu den ersten n Primzahlen teilerfremd sind","volume":"32","author":"Backlund","year":"1929","journal-title":"Ann. Acad. Sci. Fenn."},{"key":"ref_3","first-page":"116","article-title":"\u00dcber eine zahlentheoretische Behauptung von Legendre","volume":"29","author":"Brauer","year":"1930","journal-title":"Jber. Berliner Math. Ges."},{"key":"ref_4","first-page":"124","article-title":"On the difference of consecutive primes","volume":"6","year":"1935","journal-title":"Quart. J. Math. Oxford Ser. Ser."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1112\/jlms\/s1-13.4.242","article-title":"The difference between consecutive prime numbers","volume":"13","author":"Rankin","year":"1938","journal-title":"J. Lond. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1090\/S0002-9947-1990-0972703-X","article-title":"Unusually large gaps between consecutive primes","volume":"322","author":"Maier","year":"1990","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1017\/S0013091500025633","article-title":"The difference between consecutive prime numbers V","volume":"13","author":"Rankin","year":"1962\/1963","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF01234916","article-title":"Eine Bemerkung zur Konstruktion grosser Primzahll\u00fccken","volume":"14","year":"1963","journal-title":"Arch. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1006\/jnth.1997.2081","article-title":"Very large gaps between consecutive primes","volume":"63","author":"Pintz","year":"1997","journal-title":"J. Number Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"935","DOI":"10.4007\/annals.2016.183.3.4","article-title":"Large gaps between consecutive prime numbers","volume":"183","author":"Ford","year":"2016","journal-title":"Ann. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"915","DOI":"10.4007\/annals.2016.183.3.3","article-title":"Large gaps between primes","volume":"183","author":"Maynard","year":"2016","journal-title":"Ann. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.4007\/annals.2010.171.1753","article-title":"Linear equations in primes","volume":"171","author":"Green","year":"2010","journal-title":"Ann. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"465","DOI":"10.4007\/annals.2012.175.2.2","article-title":"The quantitative behaviour of polynomial orbits on nilmanifolds","volume":"175","author":"Green","year":"2012","journal-title":"Ann. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.4007\/annals.2012.176.2.11","article-title":"An inverse theorem for the Gowers Us+1[N]-norm","volume":"112","author":"Green","year":"2012","journal-title":"Ann. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1112\/S0010437X16007296","article-title":"Dense clusters of primes in subsets","volume":"152","author":"Maynard","year":"2016","journal-title":"Compos. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"819","DOI":"10.4007\/annals.2009.170.819","article-title":"Primes in Tuples I","volume":"170","author":"Goldston","year":"2009","journal-title":"Ann. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11511-010-0044-9","article-title":"Primes in Tuples II","volume":"204","author":"Goldston","year":"2010","journal-title":"Acta Math."},{"unstructured":"Goldston, D.A., Pintz, J., and Yildirim, C.Y. (2016). The path to recent progress on small gaps between primes. arXiv, arXiv:math\/0512436v2.","key":"ref_18"},{"doi-asserted-by":"crossref","unstructured":"Ford, K., Green, B.J., Konyagin, S., Maynard, J., and Tao, T. (2016). Large gaps between primes. arXiv.","key":"ref_19","DOI":"10.1090\/jams\/876"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2659","DOI":"10.1016\/j.jfa.2016.08.014","article-title":"Large gaps between consecutive prime numbers containing perfect k-th powers of prime numbers","volume":"272","author":"Maier","year":"2017","journal-title":"J. Funct. Anal."},{"doi-asserted-by":"crossref","unstructured":"Ford, K., Heath-Brown, D.R., and Konyagin, S. (2015). Large gaps between consecutive prime numbers containing perfect powers. Analytic Number Theory, Springer. Honor of Helmut Maier\u2019s 60th Birthday.","key":"ref_21","DOI":"10.1007\/978-3-319-22240-0_5"},{"key":"ref_22","first-page":"438","article-title":"The difference of Consecutive Primes","volume":"6","year":"1940","journal-title":"Duke Math. J."},{"key":"ref_23","first-page":"50","article-title":"On the number of positive integers \u2264x and free of prime factors \u2265y","volume":"13","year":"1951","journal-title":"Indag. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/0097-3165(89)90074-5","article-title":"Asymptotic behavior of the chromatic index for hypergraphs","volume":"51","author":"Pippenger","year":"1989","journal-title":"J. Combin. Theory Ser. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/S0195-6698(85)80023-8","article-title":"On a packing and covering problem","volume":"6","year":"1985","journal-title":"Eur. J. Comb."},{"key":"ref_26","first-page":"1","article-title":"Small differences between prime numbers","volume":"293","author":"Bombieri","year":"1966","journal-title":"Proc. R. Soc. London. Ser. A Math. Phys. Sci."},{"unstructured":"Davenport, H. (2000). Multiplicative Number Theory, Springer. [3rd ed.]. Graduate Texts in Mathematics.","key":"ref_27"},{"doi-asserted-by":"crossref","unstructured":"Broughan, K. (2021). Bounded Gaps Between Primes: The Epic Breakthroughs of the Early Twenty-First Century, Cambridge University Press.","key":"ref_28","DOI":"10.1017\/9781108872201"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"383","DOI":"10.4007\/annals.2015.181.1.7","article-title":"Small gaps between primes","volume":"181","author":"Maynard","year":"2015","journal-title":"Ann. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.4007\/annals.2014.179.3.7","article-title":"Bounded gaps between primes","volume":"179","author":"Zhang","year":"2014","journal-title":"Ann. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0001-8708(81)90003-7","article-title":"Chains of large gaps between consecutive primes","volume":"39","author":"Maier","year":"1981","journal-title":"Adv. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/BF01403187","article-title":"A large sieve density estimate near \u03c3=1","volume":"11","author":"Gallagher","year":"1970","journal-title":"Invent. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1134\/S0001434610090312","article-title":"Character sums over shifted primes","volume":"88","author":"Friedlander","year":"2010","journal-title":"Math. Not."},{"doi-asserted-by":"crossref","unstructured":"Maier, H., and Rassias, M.T. (2020). Prime avoidance property of k-th powers of prime numbers with Beatty sequences, In Discrete Mathematics and Applications, Springer.","key":"ref_34","DOI":"10.1007\/978-3-030-55857-4_15"},{"unstructured":"Maier, H., and Rassias, M.T. (2023). Prime Avoidance Property of k-th Powers of Piatetski\u2013Shapiro Primes. arXiv.","key":"ref_35"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"37","DOI":"10.4064\/aa157-1-3","article-title":"Piatetski\u2013Shapiro sequences","volume":"157","author":"Baker","year":"2013","journal-title":"Acta Arith."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:48:33Z","timestamp":1760028513000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,6]]},"references-count":36,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030198"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030198","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,6]]}}}