All Random Solved Random Open
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.
  • 442 Dense sets of natural numbers with unusually large least common multiples - Tao arXiv (08/07/24)
  • 756 A problem of Erdős about rich distances - Bhowmick arXiv (02/07/24)
  • 475 Rearranging small sets for distinct partial sums - Kravitz arXiv (01/07/24)
  • 265 On simultaneous rationality of two Ahmes series - Kovač arXiv (26/06/24)
  • 206 On eventually greedy best underapproximations by Egyptian fractions - Kovač arXiv (12/06/24)
  • 314 On differences of two harmonic numbers - Lim and Steinerberger arXiv (31/05/24)
  • 429 Sparse Admissible Sets and a Problem of Erdős and Graham - Weisenberg arXiv (20/05/24)
  • 121 On product representations of squares - Tao arXiv (19/05/24)
  • 29 An explicit economical additive basis - Jain, Pham, Sawhney, and Zakharov arXiv (15/05/24)
  • 268 On the set of points represented by harmonic subseries - Kovać arXiv (13/05/24)
  • 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)
  • 47 294 297 300 305 310 On further questions regarding unit fractions - Liu and Sawhney arXiv (10/04/24)
  • 297 On a problem involving unit fractions - Steinerberger arXiv (25/03/24)
  • 3 Improved Bounds for Szemerédi's Theorem - Leng, Sah, and Sawhney arXiv (28/02/24)
  • 208 Squarefree numbers in short intervals - Pandey arXiv (25/01/24)
  • 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)

(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.)