Assumption University of Windsor sponsored a symposium for mathematicians from Ontario, Michigan, and Indiana, The symposium gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 that n(2x) < 257(x) for sufficiently large x. Is it true that (1) n(x + y) < B(X) + n(y) ?-Ungrir has verified the inequality for y 5 41. Hardy and Littlewood [29, p. 691 have proved that (2) P(X + Y)-n(x) < cy/log y. In the same paper, they discuss many interesting conjectures. They put h-0 sup [R (x + Y)-n(x)1 = p(y) , x-00 and they conjecture that p(y) > y/log y, and that perhaps n(y)-p(y)-oo as y-) 00. Hardy and Littlewood deduce (2) by Bruds method. A very difficult conjecture, weaker than (1) but much stronger than (2), is that corresponding to each s > 0 there exists a yE such that, for y > yE, dx + Y)-dy) < (1 + dy/log y-It has not yet been disproved that p(y) = 1 for all y. If p(y) > 1 for some y, then lim inf (h+l-pn) < 00. 2. About seventy years ago, Piltz [38] conjectured that, for each c > 0, pn+l-pn= O(n&). Cram& conjectured [7, p. 241 that pn+l-pn = O((log n)?. If lim sup (p,+l-p,)/(log n) " = 1 , then, for each E > 0, infinitely many of the intervals [n, n + (1-s)(log n) " ] contain no primes, but for n > ns, there is a prime between n and n + (1 + E)(log n) ". The Riemann hypothesis implies that prl+1-pn < n&+1/2 [35, Vol. 1, p. 3381. Thus the old conjecture that there is always a prime between two consecutive squares already goes beyond the Riemann hypothesis.
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J. Littlewood,et al.
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1923
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A. E. Ingham.
On the difference between consecutive primes
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1937
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R. Rankin.
The Difference between Consecutive Prime Numbers, III
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1938
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P. Erdös.
Note on some elementary properties of polynomials
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1940
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The Theory of Measure in Arithmetical Semi-Groups
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1952
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1953
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Covering a Three‐Dimensional set with Sets of Smaller Diameter
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1955
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Paul Erdös,et al.
On a Problem of Additive Number Theory
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1956
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Beweis einer Vermutung von A. Vázsonyi
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1956
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On the sums of powers of complex numbers
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1956
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B. Grünbaum.
A simple proof of Borsuk's conjecture in three dimensions
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1957,
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Some problems and results in elementary number theory
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