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Category Archives: open problems
An exercise in number theory
Here is a cute little problem which I can’t solve. I thought I needed the answer but it turned out that I didn’t, so as far as I know there is no application. Let n = p1a1…psas, where the pi are distinct … Continue reading
Posted in doing mathematics, open problems
Tagged Euler's function, multinomial coefficient
6 Comments
Two things that should be related
Here are two things that look as if there should be a relation between them. A graph duality In my paper on graphs defined on groups, I invented an ad hoc duality relation between pairs of graphs. The context is … Continue reading
BCC29 at Lancaster
Last week we celebrated the 29th British Combinatorial Conference in Lancaster, face to face. (As a side observation, this was by far the largest social gathering I have been at since the start of the pandemic; I found it both … Continue reading
I’d like to see this solved
Here is a problem that I would really like to see solved. I have spent quite a bit of time on it myself, and have suggested it to a few other people, but it still resists all attacks, though it … Continue reading
Posted in doing mathematics, open problems
9 Comments
A problelm
Given a finite permutation group G on a set X, the permutation character π of G is the function on G mapping an element g to its number of fixed points in X. This is a character of G, the … Continue reading
Graphs on groups, 9
We continue to make progress with the graphs on groups project, but this post attempts to step back and look at the whole thing. What use is all this? Once, after I talked at a departmental colloquium at the University … Continue reading
Posted in doing mathematics, exposition, open problems
Tagged graph theory, group theory, number theory, open problems
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Clive Sinclair
Yesterday I read the news that Clive Sinclair has died. This brought back memories of my first encounter with personal computers nearly 40 years ago. At the time I had a demanding job and three small children, and I was … Continue reading
Posted in doing mathematics, history, open problems
Tagged Neil Calkin, sumfree sets, ZX Spectrum
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Graphs on groups, 8
The dark clouds seem to have lifted a bit. Perhaps now, that the last rush of conferences for a while is over, life can return to something like normality … For me the most significant event was the last in … Continue reading
Posted in doing mathematics, events, open problems
Tagged graphs and groups, matching number, power graph
3 Comments
A small problem
In connection with the research discussion about graphs and groups, I began to wonder which finite groups have the property that any two elements of the same order are conjugate. I thought about this, and got a certain distance, and … Continue reading
A little problem
In connection with the power graphs of unitary groups, I came across the following little numbertheoretic conundrum. Can anyone solve it? Let q be an odd power of 2 (bigger than 2). Show that (q2−q+1)/3 is not a prime power … Continue reading