The conjecture is that the outcome of the product of two numbers always is just as long as (or maybe one figure shorter than) the two numbers written one after the other.       (21.1)

For powers of ten like 10, 100, 1000, 10,000 this is obvious. E.g.

1000 (3 zeros) * 100,000 (5 zeros) = 100,000,000 (3 + 5 = 8 zeros).       (21.2)

Because of the 1's in front and the * sign, the prime product (left side of equation) when written out is two signs longer in this case than the single number of its outcome (right side of equation). Let's neglect this small inconvenience for the moment.       (21.3)

In note to page 13 are given three examples of multiplication of long prime numbers and you can count that (21.1) is satisfied in these cases.

The arguments so far suggest that (21.1) holds also for all numbers between the powers-of-ten numbers. But that are a real lot of numbers.

This is in the decimal system. One can switch to e.g. the septenary system (based on 7 instead of 10) to get smaller spacings between the powers of 10. Write out the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... up to infinity (natural numbers only). Then wipe out all numbers in which figures 7, 8 or 9 are occuring. What is left over is the septenary system: 0, 1, 2, 3, 4, 5, 6, 10, ... up to infinity. In it hold the same algebraic formulas as you know it from the decimal system. So (21.1) and (21.2) hold but now with smaller spacings between the powers of 10. Mind “10” now is the 7th number, wherein “7th” is an expression from the decimal system; the septenary system doesn't have a symbol for 7.       (21.4)

The smallest number system is the binary system. When one has put a point at the number line for all powers of ten of all number systems, one still have overlooked a real lot of numbers. For each of the skipped numbers holds (21.1), or not.       (21.5)

What I want here is a number system with a real number base instead of only the natural numbers as base. Then you can define any number you choose as being a power of 10 only. The conjecture is such a number system can exist.       (21.6)

g^10^a * g^10^b = g^10^(a+b)       (21.7)

E.g.

g^100 * g^1000 = g^100,000 (a=2 and b=3).

The general formula for any number in the g-based number system is:

... + d * g^-3 + c * g^-2 + b * g^-1 + a * g^0 + B * g^1 + C * g^2 + D * g^3 + ...       (21.8)

For g = 10 one gets the decimal system, for g = 7 one gets the septenary system, “10” and “7” to be understood in our well-known decimal system terms.

Where to get ... d, c, b, a, B, C, D, ... from? When g = 10 then e.g. d = 6, c = 3, b = 8, a = 5, B = 9, C = 9, D = 3, ... When g = 7 then b, B and C will be different figures (below 7). But when g = 2.4 then how to count from 1 up to 10?

In the decimal system the area between 0 and e.g. 1000 is divided in 10 equal pieces of 100 length. The number line is a fractal system. So the area of e.g. 2.4^3 must be able to be divided in 2.4 equal areas of 2.4^2. When starting from 0 then I can count down the first 2 parts, but when I finally want to count the last 0.4 part that part is not equal to the two others. Still the “skeleton” of the system is well visible.

How to proceed? I feel there must be something possible there. But I still didn't find out. If the g-based number system exists, any pair of numbers can be seen as powers of 10, defined in that g-based number system. Then (21.1) is proven. However, the consistency of the g-based number system still is a conjecture.