In a message dated 07/07/03 04:23:08 GMT Daylight Time,

sleephound@...
writes:

> I also see some differences from your list here:

>

> > h(-4*n) = 1 for n=1,2,3,7.

> > h(-4*n) = 2 for n=5,6,10,13,15,22,37,58.

>

> 12 and 28 don't show up for h(-d)=1.

> 60 doesn't show up for h(-d)=2.

>

> Am I doing something wrong, or so these sources disagree with your

> source?

>

h(-12) = h(-28) = 1 give n = 3 resp. 7.

h(-60) = 2 gives n = 15.

Ok?

Notice that I said "..(squarefree) n...". The list for /all/ n is a bit

longer:-

h(-4*n) = 1 for n=1,2,3,4,7.

h(-4*n) = 2 for n=5,6,8,9,10,12,13,15,16,18,22,25,28,37,58.

In fact, there's quite a serious error in my earlier claim that

> it is necessary that the class number h(-4*n) = 1 or 2, since that is the

> degree of the polynomial in the subsidiary condition

Further reading has made it clear that not only the great Gauss (1801) but

even before him the mighty Euler (c. 1750) were in posession of /65/ values of n

which have a "linear" characterisation of the type we are discussing.

To the above must be added (using our modern notation of class numbers of

fields):-

h(-4*n) = 4 for n=21,24,30,33,40,42,45,48,57,60,70,72,

78,85,88,93,102,112,130,133,177,190,232,253

h(-4*n) = 8 for n=105,120,165,168,210,240,273,280,312,330,

345,357,385,408,462,520,760.

h(-4*n) = 16 for n=840,1320,1365,1848.

[These values come from David Cox's superb book.]

Apparently it has since been proved that there is at most one other value of

n, but no-one has been able to show whether or not it exists!

This is one of the deepest and richest areas of number theory, still not

exhausted by centuries of mining.

Mike Oakes

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