Correspondence: J. C. P. Miller, Sc. D.

7 de Freville Ave

Cambridge

1974 Oct 11

Dear Mr. Anand,

Professor Cassels has shown me your paper on Two Series containing the

Primes and a Conjecture concerning the Prime Difference.

I agree with him that I cannot find anything in it of very great significance.

Nevertheless, I found it of some interest to read and to think about for a while, so

I am glad to do what Professor Cassels asked, and send my comments to you for

consideration. My notes are handwritten and rather rough, but I hope you can

read them and that you may find them useful. I apologise for mistakes in

calculations (performed mentally) – this would be avoided in automatic

calculations.

My recommendation is that you think of this paper as a recreational

development, or a way to get quickly at early primitive method in various

Arithmetic Progressions. Your extra numbers are rather trivial accidents, easily

avoided by a revision of the process to include not merely a(I, j) but a(I, j)+2t(j),

a(I, j)+4t(j) … so long as the new d(j) is not exceeded by them; if d(j) is equal to

one of them, then this d(j) is excluded. This is very easily done electronically.

As a recreational paper one might attempt to publish in an educational journal,

such as the Mathematical Gazette here, or the American Mathematical Monthly.

I cannot see that there is much to contribute to the study of the gap between

successive primes. A fair bit of calculation has been done, and it seems likely

that the biggest gaps are of order (logN)², with a much smaller average. But

nothing has been proved. However, I have not yet studied this part at all

carefully, so I will not now comment.

I send this letter, with comments, to you fairly quickly, so that you may not have

too long to wait, and because I have not time to continue this study at the

moment, much as I have enjoyed playing about with the idea.

I hope my comments are useful to you.

J. C. P. Miller

Sc. D.