◄ Index

◄ Main essay

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

1. TRIM numbers

Why not even numbers? 2⁴, 2⁶, 2⁸, … are Trim-numbers. In fact, 2ⁿ, except when

2ⁿ-1 is a Mersenne prime. Also 96=2⁵.3, and generally 2ⁿ.3 when 3.2ⁿ-1 is not a

prime. Also 120.125 i.e 2 between 113 and 127. There are seven between 1327

and 1361, 6 even and 1 odd.

120 and 125 are A₃ numbers, in the sense of Western and Miller, R. S. Math.

Tables Vol. 9.

2. Indices and Residue Indices

An A_n number has all prime factors ≤ p(n), and formulae have been obtained for

the numbers less than a given limit. These might suffice to show that, even if

even numbers are omitted, the frequency is such that 2 or more in a large

interval between primes is likely.

There has been quite a lot of study of prime gaps d(i), the largest is conjectured

of order (log_en)². I have factored some largest gaps e.g. 180 at 17051707

(I think!). I must look to see if there are enough A_n numbers, for p(n)=179, in the

neighbourhood.

3. The numbers a(i, j) can be obtained directly.

In fact, I think:

a(i, j) = t(j) - r(i, j)

where t(i) = λt(j) + r(i, j), 0 < r(i, j) =< t(j), λ an integer, and these can be obtained

directly by division.

(a(i, j) is, in fact, the difference between t(i) and the next multiple of t(j).)

Now, your recurrence neatly avoids the division but is two-dimensional, also very

time consuming in a check (?).

Nevertheless, a direct recurrence (one-dimensional) for r(i, j) is known, but uses

all odd integers; this can be used to test isolated large numbers. This is due to G.

G. Alway, Math. Tables and Other Aids to Computation (M. T. A. C) Vol 6, p.59-

60, 1952.

This method can be developed further.

4. Your omission of odd d(i) is justified

by the fact that they are all covered by:

a(i, 1)+2λ, a(i, 1) = 1 always.

Your inclusion of 27 is due to the fact that you don’t use a(i, 2)+3λ, except for

λ=0.

e.g. for t(i)=23, a(i, 2) = 1 and a(i, 2)+3 = 4, which does not otherwise occur.

Similarly for 125 your terms would be (if I’ve not erred):

…

113

…

I don’t go to the end because it must be adequate to go to 11 (as in the Compact

series) i.e. integer part of (113)^1/2, or 13, exceeding p(n)^½.

We note that 2, 4, 6, 8, 10, but not 12 or 14 are covered – though 4 and 10 rather

later. But add multiples of 3 to a(i, 2) – this covers both “late” items; now 5 to

a(i, 3) – this removes the Trim number 125.

5. To sum up

The choice of rules for TRIM-numbers seems rather artificial, but the idea of a

simple recurrence is useful, but not new. The fact that the recurrence is basically

two-dimensional is a drawback.

Nevertheless, the connection with A_n-numbers is one I find interesting, and the

queries about the occurrence of TRIM-numbers (both odd and even) are

intriguing curiosities.

6. One can easily cut out TRIM numbers

by noting the zero item, and looking for the next missing even integer in the line

above. In fact, it seems to me that one would step ahead with any “missing” step.

This might be a useful idea.

7. Better, if you get a zero in line i+1, just add p(j) if a(I, j) is odd

or 2p(j) if a(i, j) is even, and modify d(i) to be the next larger even integer. This

works for COMPACT series as well. I think you are left with primes alone.

8. I see the COMPACT series is just the same idea

but used only upto c(i)^1/2 in the columns. I think my previous comments, then,

cover this case as well.

9. However, I note that if you use only even d(i)

you should also use only even a(i, j), i.e. if it is odd, add another t(j). Of course

you must have 1 in the t(1) column.

With “even” residues you still have extra numbers, but fewer – that is, when you

only keep one residue in each column. If you add 2t(i) to the previous line when

you get zero, you can avoid them.

I find the first composite c(i)* (with even residues) is 207 which you miss in yours

in favour of 205: however we both get 219.

10. There’s quite a lot of fun to be had in these tables.

I think though that Alway’s process is more useful for an isolated test of primality.

You can also use Alway’s method to start yours at any particular place, and then

continue with yours. Always’s process must be used with equally spaced divisors

(at least it has not so far been adapted to irregular spacing). Yours just uses (or

could just use) primes.

11. Your process could be adapted easily to primes

in specific arithmetic progressions.

I give an example for 6n+1. The primes 5, 11 etc. must also be included, since

5x11 = 55 = (6x9)+1 will appear; it should either be excluded, or counted as

“primitive” i.e. not prime, but no factors of the “right” form.

All the d(i) have to be multiples of 6, and it helps to make all the a(I, j) multiples of

6. If not, the composite 259 = 7x37 seems the first to appear, but multiples of 6 in

the previous line exclude it.

12. Your series of primes with some composites reminds me

of another of some significance, and much studied. These are the Fermat

composites.

For any integer base a, there is a sequence of numbers n(i) for which:

aⁿ⁽ⁱ⁾- a ≡ 0 (mod n) … (X)

This sequence includes 3 kinds of numbers.

(i)

All primes (Fermat’s Theorem);

(ii)

Some composites, Poulet numbers, satisfying (X) for a, but not for all

bases (though each n(i) also satisfies (X) for some other bases – not

all same for all n(i));

(iii)

Some composites, Carmichael numbers which satisfy (X) for all bases a.

Poulet numbers may have 2 or more factors. Carmichael numbers have at least

3 factors.

e.g.

91 is Poulet for a=2, i.e. 2⁹¹ ≡ 2 (mod 91)

341 is Poulet for a=10, i.e. 10³⁴¹ ≡ 10 (mod 341)

561 is Carmichael, i.e. a⁵⁶¹ ≡ a (mod 561) for all a.

The non=primes are rare, but prevent the simplest use of (X) as a test for

primality. Incidentally n may be even, the last case for a = 2 is, I think, around

160000 somewhere.

We can extend the test, fortunately, to distinguish primes from composites – but

with occasional major difficulty.

J. C. P. Miller

1974 Oct 8