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Problems have always been an essential part of my mathematical life. A well-chosen problem can isolate an essential difficulty in a particular area, serving as a benchmark against which progress in this area can be measured. It might be like a 'marshmallow', serving as a tasty tidbit supplying a few moments of fleeting enjoyment. Or it might be like an 'acorn', requiring deep and subtle new insights from which a mighty oak can develop. […]

In this note I would like to describe a variety of my problems which I would classify as my favorites. Of course, I can't guarantee that they are all 'acorns', but because many have thwarted the efforts of the best mathematicians for many decades (and have often acquired a cash reward for their solutions), it may indicate that new ideas will be needed, which can, in turn, lead to more general results, and naturally, to further new problems.

In this way, the cycle of life in mathematics continues forever.

Paul Erdős, Some of my favorite problems and results, 1997
There are 808 problems in the database of which 235 (29%) have been solved.
RECENT PAPERS
  • 442 Dense sets of natural numbers with unusually large least common multiples - Tao arXiv (08/07/24)
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  • 29 An explicit economical additive basis - Jain, Pham, Sawhney, and Zakharov arXiv (15/05/24)
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  • 297 A question of Erdős and Graham on Egyptian fractions - Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte arXiv (24/04/24)
  • 585 Edge-disjoint cycles with the same vertex set - Chakraborti, Janzer, Methuku, and Montgomery arXiv (10/04/24)
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  • 3 Improved Bounds for Szemerédi's Theorem - Leng, Sah, and Sawhney arXiv (28/02/24)
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  • 213 On integer distance sets - Greenfeld, Iliopoulou, and Peluse arXiv (19/01/24)
  • 549 On the Ramsey number of the double star - Dubó and Stein arXiv (02/01/24)
OPEN -> SOLVED

(Some are new solutions, some are old solutions new to this site. The date is when the site was updated with the change of status.)