Peter Cameron's Blog

As you know, this is where I take out my frustrations by having a good grumble about life, the universe, and everything. Yesterday, in a meeting, we learned of two developments coming to the academic world. In neither case is it an unmixed blessing. I should add that this is in no way a grumble against my colleagues, who foresightedly and openmindedly have told us about what is heading our way so that we can be prepared.

The first, and mostly harmless, concerns publication of research data. No longer will it be enough to say “Contact the author for the data/programs used in this paper”; the data must be available at a specific link published in the paper. It is even claimed that such statements will still be required in papers with no data at all, such as pure mathematics papers.

I do not know whether this is being forced on the humanities as well as the sciences, or whether it is just an anomaly of mathematics being misclassified as a science. In any case, I hope that someone can provide us with some boilerplate text for the purpose. (One of my colleagues suggested something along the lines “No data were hurt in the research underlying this pure mathematics”.)

But how do you publish “no data”? If mathematics is founded on set theory, a file containing the empty set symbol might do the job (an empty file might be problematic). But perhaps the bureaucrats might be happier with a spreadsheet with no entries.

Journals already require publication of funding information; many specify a conflict of interest statement; some need an ethical approval statement. I fear that in future a pure mathematics publication will resemble a beautiful work of art in a standard frame designed by a committee.

The second, more serious because it addresses a real problem, concerns accessibility of lecture notes. Now this doesn’t mean, as you might first think, that we will have to dumb down our lecture notes to make them accessible to creatures of restricted intelligence. Instead, it seems that there is a PDF reader program which reads aloud the contents of a PDF file for disabled students. (No, I am not talking about a lecturer!) It seems that PDF documents produced via LaTeX are not well handled by this piece of software, and so we are to blame and must suffer.

I have spent a lot of time during my career trying to make lecture notes as clear and beautiful as possible. I have always taken the view of conventional typographers, that typesetting is a window through which you see the beautiful scenery, and the best typesetting is the one which provides the most unobtrusive window pane. Now we are told that the programs to fix this problem (the best of which is called “bookdown”, if I caught the name right) seriously degrade the LaTeX typesetting in the interests of accessibility. So the majority of students will be penalised to help the minority. Is this good? This is a moral question I can’t judge.

With a bit less than twice the work, one could produce two files, one a carefully crafted LaTeX file, the other a reader-friendly file. The downside would be maintenance; changes would have to be made in two places, and the files could very easily get out of sync.

Let me interpolate two speculations here; I have no evidence for either of these.

First, it seems highly likely that this PDF reader is optimized for files produced by Word, as a hangover of the Microsoft hegemony. If this so, then we are being punished for using a better product than Microsoft can produce.

Second, this measure (entirely arbitrary, if my first speculation is correct) will delight the bureaucrats. Here is a measure, a number produced by computer and hence completely objective, of our concern for handicapped students. How long until it is incorporated into staff appraisals and disciplinary measures?

Now I have a suggestion here. The great computer scientist Donald Knuth, in the days before the term “web” became synonymous with “internet” in the public mind, devised a system for producing programs and documentation together, which he called Web. A Web document could be fed to two preprocessors called Tangle and Weave, though I don’t remember which was which. One of them produced as output a Pascal program (Knuth’s preferred language at the time; I believe there is a C version of Web now). The other output a TeX document which consisted of the program with documentation. Knuth published two major programs, TeX and METAFONT, in this form as books.

Surely we could have a much simpler system here. An input file (which would superficially resemble LaTeX, to ease the learning curve) could be processed in two ways: the output of one would be a reader-friendly PDF, or means to produce one, such as a bookdown file; the other would be a non-crippled LaTeX file. Both could be made available to students, and maintenance would only need to be done in one place.

Ultimately, like Hamlet, I know where the exit door is, and I know it is not locked. When my previous university brought in draconian rules for staff performance and appraisal, including restrictions on seminar attendance, I decided that rather than stay and fight I would retire and live on my pension; and so I would be doing, had not St Andrews come to the rescue with the offer of a half-time position. I still have the option of retiring and living on my pension. But I am in such a good, friendly and supportive department that I don’t expect this to become necessary, and I would take that door only with great reluctance.

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 of Southern California, after my talk someone asked “What use is all this?” My view is that the fact that it is fun is reason enough to do it. But people like funders may want more than that as an answer. So let me make some comments.

To group theorists

A group theorist is likely to ask: What do I learn about a group from these graphs? So I will address that first.

The subject began with the seminal paper of Brauer and Fowler in 1955, in which they showed that there are only finitely many finite simple groups having a given group as the centralizer of an involution. Of course, it wasn’t then known that a non-abelian finite simple group must contain an involution; at the time, this was “Burnside’s conjecture”, and was proved by Feit and Thompson in the next decade. After Brauer–Fowler and Feit–Thompson, it was clear that characterizing simple groups by the centralizer of an involution would be an important ingredient in any eventual classification; and so indeed it turned out.

Brauer and Fowler used the reduced commuting graph (the commuting graph with the identity removed). I think the word “graph” does not occur in their paper, but the make use of the graph distance. Essentially they showed that the distance between two involutions in this graph is rather small.

Maybe the next contribution was that of Gruenberg and Kegel, who defined the prime graph of a finite group (now called the Gruenberg–Kegel graph: the vertices are the prime divisors of the group order, two vertices p and q joined if there is an element of order pq in the group. They gave a powerful structure theorem for groups whose GK graph is disconnected. Moreover, this was a tool in studying the decomposition of the augmentation ideal of the integral group ring.

I will return to this question later.

Another thing of interest to group theorists is that questions about the graphs often give rise to challenging problems about the groups. Here are three examples.

1. The following three properties of a finite group G are equivalent:
   • G is an EPPO group (all elements have prime power order);
   • the Gruenberg–Kegel graph of G has no edges;
   • the power graph and enhanced power graph of G coincide.
   So we have the problem of classifying these groups.
2. Other questions about when two of these graphs coincide give rise to other group-theoretic classification problems, some of which are worth a look. Examples include minimal non-abelian, non-nilpotent, or non-solvable groups, 2-Engel groups, and Dedekind groups.

To graph theorists

By contrast, graph theorists are not going to learn anything about general graphs by studying graphs derived from groups. However, there is a different motivation, cited by Kelarev and Quinn when they introduced the power graph in around 2000: Do groups give us examples of graphs with interesting properties, for example, graphs which will be good as networks (so good connectivity and expansion properties)?

In looking for a good network, there is a tension between having many edges (which will make the network more expensive to construct) and having good connectivity (too few edges work against this). I don’t know any definitive result. But when I was involved in writing a survey paper on power graphs of groups, as an experiment I looked at an example, the Mathieu group M11, the smallest sporadic simple group. This group has order 7920, so we start with a graph with 7920 vertices. But simple reductions (first “twin reduction”, identifying vertices with the same open or closed neighbourhood, then discarding “peripheral” parts of the graph) results in a rather beautiful graph: a bipartite graph on 165+220 vertices, which is semiregular (valencies 4 and 3 in the two parts), has diameter and girth 10, and has automorphism group M11. I have not examined this graph further, and have not done a thorough search to see if it has been found before, but finding something as nice as this in the first place I looked seemed a good omen.

To other mathematicians

I can offer here, as a carrot, the fact that this research has thrown up a number of interesting diophantine problems in number theory.

For one example among many, the power graph and enhanced power graph of a group have the same clique number if and only if the largest order of an element of the group is a prime power. I do not expect that much can be said about this class of groups. But let us look at one example, the groups PGL(2,q) for prime powers q. The largest order of an element in this group is q+1. So the power graph and enhanced power graph have the same clique number if and only if either

• q is a power of 2, and q+1 is a Fermat prime or 9; or
• q is a Mersenne prime, and q+1 is a power of 2.

(The first bullet point here conceals the use of the solution to Catalan’s equation, prime powers differing by 1.)

Other number-theoretic problems are thrown up by other questions. But here is one which is unrelated.

A clique in the power graph of G is contained in a cyclic subgroup of G. So the clique number of the power graph of G is equal to the largest clique number of a cyclic subgroup of G. (This is not the same as the clique number of the largest cyclic subgroup.) Now the clique number of a cyclic group of order n is a number-theoretic function f which is defined by the recurrence f = 1, and for n > 1, f(n) = φ(n)+f(n/p), where p is the smallest prime divisor of n, and φ is Euler’s totient. It follows that f(n)<3φ(n) for all n. But we could ask:

• Is the lim sup of the ratios f(n)/φ(n) rational, algebraic, or transcendental?
• What other limit points does the set of these ratios have?

To young researchers

Finite group theory is a very technical area, often somewhat off-putting to outsiders. But graph theory is vast and sprawling with many easy ways in.

So, not surprisingly, the field of graphs on groups offers many problems where mathematicians starting out can work profitably.

For example, there are many graphs defined on groups: power graph, enhanced power graph, deep commuting graph, commuting graph, nilpotence graph, solubility graph, non-generating graph. In a paper in preparation, we are going to extend this hierarchy into a second dimension. And there are many graph properties and parameters: connectedness, diameter, girth, clique number, chromatic number, independence number, clique cover number, matching number, domination number (strong or weak), Eulerian, Hamiltonian, pancyclic, and lots more.

So choose a type of graph on a group, and a property or parameter; chances are reasonably good that this will not previously have been investigated. See what you can do!

There are other areas of algebra where similar graphs arise, and where similar questions can be asked. One of these is zero-divisor graphs of rings (that is, commutative rings with identity).

Some of my favourite problems

Since we have a hierarchy, which is being extended into a second dimension, one very natural thing is to compare two graphs in the hierarchy. Most questions about the original hierarchy have been solved, and interesting graph classes arise. But there will soon be more such questions.

Also, even if these problems have been solved, they can be generalized in the following way. Take two adjacent graphs in the hierarchy, call them A and B; and take a monotone graph parameter p, one which cannot decrease when an edge is added (or one which cannot increase, if you prefer): many parameters are monotone. Now ask: For which groups G does p(A(G)) = p(B(G))? Two small examples: the clique numbers of the power graph and enhanced power graph of G are equal if and only if the largest order of an element of G is a prime power; and the matching numbers of these two graphs are equal for all G (even though we cannot write down a formula for them in every case).

Another problem area concerns universality. Given a graph type A, can every finite graph be embedded as induced subgraph in A(G) for some group G? If not, can you identify the class of graphs which can be so embedded? And in the other direction, given a subgroup-closed class of graphs, can you identify the class of groups G for which A(G) belongs to that class?

If I could send a message to the world leaders who will soon assemble in Glasgow, it would be, in the words of a St Andrews honorary graduate:

What’ll you do now, my blue-eyed son?
What’ll you do now, my darling young one?

You have been around the world, seen and heard the evidence (the scientific data, the fires, the floods), and met the people whose lives will be most affected; now is the time to act before we “start sinking”.

In fact this song, written nearly 60 years ago, is full of topical references to the state of our world. For example, “sad forests” (acid rain), “dead oceans” (plastic pollution), “graveyard” (Covid), “black branch” (destruction of forests), “white ladder covered with water” (sea-level rise), “guns … in the hands of young children” (civil wars in Uganda and elsewhere), “wave” (from melting icecaps? or a tsunami?), “one man starve” (famine), “song of a poet” (here it is!), “young woman whose body was burning” (the next pandemic?), “man who was wounded in love/hatred” (persecution based on sexual orientation in some countries, the burden of prejudice that so many people have to carry), and so on.

Perhaps the song should be the anthem of COP26?

Another sad loss, with the death of Joachim Neubüser this week.

He was the driving force behind the creation of the computer algebra system GAP. Even though I worked with John Cannon, the creator of the rival system Magma, I have always used GAP rather than Magma. That for several reasons:

• Magma is expensive, GAP is free, and I am mean. Neubüser is supposed to have said, “Nobody charges you for using Sylow’s Theorem, so nobody should charge you to compute a Sylow subgroup.”
• It had unlimited precision integer and rational arithmetic, which was absolutely necessary for some of the computatations I did – indeed, before I started using GAP, I had written my own unlimited precision integer routines in Pascal. Along with everything else, I use GAP as a calculator.
• It happened that I have always had GAP expertise close at hand: Leonard Soicher at QMUL, Alexander Konovalov and a number of others at St Andrews. Indeed, Leonard wrote the GRAPE package for computing with graphs and groups, which I have made huge use of.

Related to the second point, back in 2008 I spent six months in Cambridge running a programme at the Isaac Newton Institute. Jan Saxl very kindly arranged for me to be a visiting fellow at Gonville and Caius College, which gave me the use of a flat in Rose Crescent. At the time Rosemary was recovering from cancer treatment, so it was possible to come to Cambridge and spend her time in the flat apart from short walks. She had just been on an RSS panel investigating the distastrous failure of a drug trial at Northwick Park, and had invented some better designs for first-in-human drug trials. To find out how much better than the designs then in use, she needed to do a lot of boring calculations involving finding the Moore–Penrose inverses of matrices. She needed the arithmetic to be exact, so GAP was ideal for the job; I set her up with a laptop and she could do the sums. (These designs have been published but never implemented despite their advantages: they would need to be approved by a regulator, and getting approval costs serious money; drugs companies would not pay, since this would be tantamount to admitting that their existing designs are less than perfect. So it goes.)

Neubüser was much more than just the originator of GAP: a skilled computational group theorist himself, and an inspiring teacher. Indeed, when I wrote my book on permutation groups in the late 1990s, he wrote me a long account of the trials and mis-steps in the classification of transitive subgroups of low-degree symmetric groups, and permitted me to quote it in the book (which I did).

But GAP, together with his students, make up his main legacy.

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 not going to get something like a Commodore PET as some of my colleagues did. Instead, I started on a Sinclair ZX Spectrum in 1982 or 1983.

The machine had a laughably small amount of memory by today’s standard: 16Kb ROM, 48Kb RAM (of which nearly 7Kb were taken up by screen memory and printer buffer, leaving only 41Kb for the user). The machine had its own version of BASIC built-in, with keywords available by a single keypress. But it used a Z80 processor, a variant of the Intel 8080, so with a list of Z80 opcodes it was possible to program it in machine code, by writing a BASIC program to put the appropriate values into memory and then call the code; it was even possible to return a value.

So I used it to prove a theorem.

Suppose that you choose a sum-free set S of natural numbers by the following random procedure. Examine the natural numbers in turn. When examining n, if it happens that n = x+y where x and y have already been put into S, then obviously n cannot be in S; otherwise, choose randomly and independently with probability 1/2 whether to put n into S or not.

This is a fascinating process with many curious properties. It is very easy to see that the probability that no odd number is ever put into S is zero. (If no odd number has so far been put into S, then the next odd number to be examined will have even chance of being put into S.) What about even numbers?

Theorem In the above process, the probability of the event that S contains no even numbers is non-zero.

In fact this probability is about 0.218.

So how does the proof go? Let pn be the probability that no even number occurs when the numbers 1,…,n have been considered; then the sequence (pn) is decreasing, and the probability p that there are no even numbers in the final set is its limit. Now there is a simple upper bound for pn−p, so if we can find a value of n for which pn exceeds this bound then the theorem is proved. The value of pn for n even can be computed by finding, for all sets of odd numbers in [1,…,n], the probability of obtaining this set as the initial part of S; a simple sum over 2n/2 subsets. A slow process, but just the job for the computer. With about a day’s computation on the Spectrum I was able to find a lower bound of about 0.21759 and an upper bound of about 0.21862 for this number.

At various times since, I have thought that it would be possible to get a more accurate estimate; computers today are faster, and the algorithm is easily parallelisable (the individual terms in the sum can be computed independently). But I never got round to it.

Using this, and a similar estimate for a kind of “cross-sum”, I was able to show that, if T is a complete sum-free set modulo m (that is, it is sum-free in the integers mod m, and every element of this quotient not in T is the sum of two elements in T), then the probability that a random sum-free set is contained in the union of congruence classes given by T is positive. (The class {1} modulo 2 is the first example of such a set.) So, unlike some of my favourite examples like the random graph, this measure space contains infinitely many “natural” pieces with positive probability.

And there is more. After a hectic week visiting Neil Calkin in Atlanta, during which we changed our mind several times about whether we were proving that it was zero or positive, we showed that sum-free sets in which 2 is the only even number have positive probability (though rather small). More generally, a set which belongs to T mod m (as above) with finitely many exceptions has positive probability. So there are more such pieces.

I’t like to advertise some open problems about this space, involving the density of a sum-free set (a random variable on the space).

• Does a sum-free set have a density with probability 1?
• Is it true that the spectrum of densities is discrete above 1/6?
• Does it have a continuous part below 1/6?

We think that the answers to the first two questions is “yes”, but are quite uncertain about the third.

By the end of the decade, I had moved to an Atari ST, which could store information on floppy discs. I was commuting from Oxford to London by then, and got a Tandy TRS-80 for writing on the train; I had to write a low-level program to connect it to the Atari, which involved learning about DTR and CTS and other mysteries of RS-232 to do this. By the following decade, though, all these things were obsolete and we were provided with PCs running some version of Windows, which crashed so often that as soon as I could get something better, I did.

But it was Clive Sinclair’s cheap and cheerful machine that got me into this.

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 the series of research discussions on graphs and groups, run by Vijayakumar Ambat and Aparna Lakshmanan S. in CUSAT, Kochi. The whole series was a great success. I gave the last talk, and devoted it to describing five areas, mostly involving the power graph of a group, where significant progress has resulted, usually as a result of contacts made during the RDGG meetings. In fact, almost as soon as I had given the talk, it was outdated: Swathi V V and I had answered, in a way which came as a surprise to me, one of the questions I asked in my talk. (This is described below.) Anyway, as well as getting the slides from the usual place on my St Andrews web page, you can watch videos of any or all of the talks in the discussion group. The link is https://youtube.com/playlist?list=PLemEI0kNLyjsrc95V78CQs3go-1sPS15T.

Another very nice meeting that I didn’t have the time or energy to comment on before was the 80th meeting of the Portuguese Mathematical Society. I was invited to speak, and chose a topic which I hoped would be accessible to many participants: “Conversations between graphs and groups”. I chose four areas where these two rather different parts of mathematics have interacted fruitfully. Again, the slides are in the usual place, and the professionally produced video is available from https://m.youtube.com/watch?v=OlELf0jBpPM&feature=share.

Anyway, here is a description of the new result. I have been banging on for a while about a hierarchy of graphs whose vertex set is a given group. I will only need two of them here:

• the power graph, in which x and y are joined if and only if one is a power of the other;
• the enhanced power graph, in which x and y are joined if they are both powers of some element z (equivalently, if the group they generate is cyclic).

Obviously any edge of the power graph is an edge of the enhanced power graph; said otherwise, the power graph is a spanning subgraph of the enhanced power graph. The two graphs coincide if and only if the group has the property that every element has prime power order. These are the so-called EPPO groups, which have been classified.

As a generalization of the classification of EPPO groups, I posed the following general problem. Let f be a graph parameter which is monotone, in the sense that adding edges to a graph doesn’t decrease its value. Then the value of f on the power graph of a group G is less than or equal to its value on the enhanced power graph. If these two graphs coincide, the values will be equal. But easy examples show that they may be equal for a wider class of groups. So I proposed the question: given a monotone graph parameter f, for which groups G do its values on the power graph and enhanced power graph coincide? This will be a wider class than the class of EPPO groups, and of course it depends on the chosen parameter.

An example of a monotone graph parameter is the matching number μ(Γ), defined to be the largest cardinality of a matching, a set of pairwise disjoint edges of the graph. Now our theorem says the following: the class of finite groups for which the power graph and enhanced power graph have equal matching number is the class of all finite groups!

Here is a hint at the proof. Take a matching M of maximum cardinalitly in the enhanced power graph of a group G. If all edges in M belong to the power graph, we are done; so suppose not. We are going to replace M by another matching of the same size, with one fewer edge not belonging to the power graph; after finitely many such steps we will be done.

Choose an edge {g,h} which is in M but is not an edge of the power graph, and (subject to this) the least common multiple of the orders of g and h is as large as possible. This lcm, call it l, is equal to the order of the cyclic group C generated by g and h. Now C has φ(l) generators, say x1,…,xφ(l), each of which is a dominating vertex of the induced subgraph on C, that is, joined to all other vertices. (Here φ is Euler’s totient.) If one of these vertices, say xi, is not covered by M, then we can replace the edge {g,h} by the edge {g,xi), which belongs to the power graph; so we can assume that there are elements y1,…yφ(l) such that, for all i, {xi,yi} is in the matching M.

Now there are three possibilities for such an edge:

• xi; is a power of yi;
• yi is a power of xi;
• neither of the above.

In the first case, g and h are powers of yi, so we can replace the edges {g,h} and {xi,yi} by {g,xi} and {h,yi}, both in the power graph. In the third case, the least common multiple of the orders of xi and yi is greater than l, contradicting the choice of {g,h}.

So the second case must apply for all values of i, and the vertices y1,…yφ(l) belong to C. If any two of them, say yi and yj, are joined in the power graph, then we can do a three-way swap, replacing {g,h}, {xi,yi} and {xj,yj} by {g,xi}, {h,xj} and {yi,yj}.

So we have an independent set of size φ(l) in the cyclic group of order l. But, using elementary number theory, it can be shown that this is only possible for l ∈ {2,6}, and these cases are easily dealt with.

So, a quick question: Let φ be Euler’s function and τ the divisor function. Then φ(n) > τ(n) unless n is one of the numbers 1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30. Is this written down anywhere? (If you haven’t seen it, it is an interesting exercise.)

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 then other jobs crowded it out of my thoughts.

I had become convinced that there are only three such groups, the symmetric groups of degrees 1, 2 and 3, and had got part way to a proof. In the end, not having time to go back to it, I resorted to a well-known search engine. With a little further research I came up with the answer, and with a small problem, which I want to pose.

In 1933, G. A. Miller published a paper in the Proceedings of the National Academy of Sciences of the USA on the problem. (His name is one which has come up a number of times in our researches about graphs on groups.) He showed, in brief, that in a group with this property, the derived group has index at most 2; moreover, if it has index 2, then the group is the symmetric group of degree 2 or 3. He couldn’t deal with the remaining case (though he did reduce it to considering non-abelian simple groups), and the earliest solution of the problem I found was by Patrick Fitzpatrick, in the Proceedings of the Royal Irish Academy in 1985; he used the Classification of Finite Simple Groups to show that the only such group is the trivial group. This was proved again by Walter Feit and Gary Seitz in 1988. (As a student of Peter Neumann, I know better than to say that they reproved the result – he would certainly have reproved me for saying so!)

So the problem I wish to pose is to give a proof of this simple result which does not depend on CFSG.

Apologies for this. If you don’t like reading personal news, especially bad news, don’t persist. I am writing it down in the hope of putting it behind me.

Back in early April, Rosemary had a very bad fall on the Fife Coastal Path. She tumbled over a six-metre cliff into a rocky river bed, full of icy water. I was able to lift her head out of the water, but it was an hour before the ambulance and coastguard were able to get her out of the river on a scoop and into a helicopter to take her to Ninewells Hospital in Dundee. She had multiple fractures to vertebrae and pelvis as well as scalp lacerations, and her right eye had moved out of position. She was in hospital for two months. She is home now, but I am her full-time carer. I don’t mind doing this, but it eats up time and energy, and I know I am not coping well with other things; I try to get some things done but there is no way I can deal with everything that arrives.

The medics expect her to make a full recovery. She can now walk several blocks on elbow crutches, and the eye is moving back into its correct position. But the vertebrae have not yet mended, so she still has intermittent pain. Part of my job is administering painkillers four times a day. Of course I have all the cooking, cleaning, laundry, and so on, as well as helping her wash and dress herself. It will be months rather than weeks before she is recovered, so I will be on the treadmill for a while yet.

I know this is all very mild compared to what she has been through, but one of the worst things I have to deal with is the memory of watching her fall and being completely unable to help. In the first weeks after the accident it took all my mental strength to stop this scene replaying over and over. I am coming to terms with it a bit better now.

So finally an apology for the rather intermittent updating of this blog. I have been to two outstanding on-line conferences over the past couple of weeks: the British Combinatorial Conference and the Portuguese Mathematical Society’s 80th anniversary meeting. Normally I would have reported on some of the wonderful things I learned at these. But because of circumstances I had to skip many conference talks that I would have liked to attend, and have not had time or energy to write about the ones I did hear. I will just say that, at the Portuguese meeting, several plenary talks which appeared at first sight to be some way from my interests turned out to be absolutely fascinating. The BCC plenary talks have already been posted and the Portuguese talks will appear soon, I hope. When this happens I might try to point you to some of my favourites.

In connection with the power graphs of unitary groups, I came across the following little number-theoretic 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 unless it is a prime.

This is an appeal for help. Has anyone come across the constant 2.648102…?

Here is the background, which connects with my previous posts about graphs on groups. We are interested in the clique number of the power graph of the cyclic group of order n. The generators of the group are joined to everything else; there are φ(n) of them, where φ is Euler’s function. These must be in every maximal clique. To them, we adjoin a maximum-size clique of the subgroup of order n/p, where p is the smallest prime divisor of n. Thus the size f(n) of the largest clique satisfies the recurrence f(1) = 1, f(n) = φ(n)+f(n/p), where as above p is the smallest prime divisor of n. (This was proved by Alireza, Erfanian and Abbas in 2015.)

Now, with f defined as above, I conjecture that the ratio f(n)/φ(n) is bounded, and its lim sup is the constant 2.648102….

As you might expect, the largest values of this ratio are realised by the product of the first so many primes.