I do not recall the rationale, that this anonymous guy "Archimedes Plutonium" (AP) led to his long term discussion, although I tried to follow his postings for a certain time. But I found a parallele to a related game, that I had started as well a certain time ago. That was playing around with constructions like "...3333" or "...147826147826". Today I do not really remember, what my idea was with constructions like that. It was, when I reviewed my understanding of the irrationals and transcendents - possibly just as a mirroring of the periodic or aperriodic decimals in front of the decimal-stop.

Now - AP repeatedly claims, that with adics, as a new representation of infinite numbers ( don’t know whether he also implies "actual infinite values") he can prove, that "Fermat’s last Theorem" (FLT) is not valid, if you use the adics construction for numbers. As I point out, there are two fundamental problem in AP’s articles: Fermat was recurring on positive integers for a,b,c when he said, that a^x + b^x <> c^x if x element of {3,4,5....}, and he did not say, that constructions like the adics of AP behave like that. Second: AP claims for his adics, that they could represent a superset of naturals, or better, they are the true naturals, and the common so called naturals just a crooked subset. As I try to show also this idea is not very likely; with more right you relate the rational numbers between 0 and -1 to the adics, and so theses concerning adics are more likely to meet theses concerning this rationals instead concerning natural numbers.

So all the mostly vehement attacks addressed to Wiley, whose proof was widely discussed last years, go simply where no one really stands.

The construction of the adics are that of a certain type of divergent series. It’s for example

â = ....1111111

which means that

â = sum ( 1*10^0 + 1*10^1 + 1*10^2 + ... + 1*10^i ...) for i=0 to infinity.

To a divergent series a certain value cannot be attached, and the question is, whether it makes any sense at all to try to deal with this series. Immediately is is obvious, that adics are not that, what Fermat meant in his statement: integers have an actual value.

Divergent series have more problematic properties; even simple reorderings can suggest completely different characteristics of the whole series, as the following example shows:

1 - 2 + 3 - 4 + 5 - 6 + 7 .... -> ???

(1-2) + (3-4) + (5-6) .... -> (-1) + (-1) + (-1) ... -> - infinity

1 -(2-3) - (4-5) ... -> (1) - (-1) - (-1) ... -> + infinity

This example also illustrates, that we obviously cannot assign any value to the series.

The scheme of the adics construction however shows, that it is a special divergent series, which has strong restrictions, and possibly by that means they allow to deal with the series in a senseful way.

• First, there are no alternating signs
• second, they are polynoms, where all sequence elements have the same basis, and their digits represent the coefficients of the polynom. Note, that the coefficients are always thought to be "normalized", i.e. smaller than the base, to behave as digits of a system.
• third, due to the writing as a decimal number there is a strong focus on the index over the sequence-elements, and for addition, multiplication etc there is always defined by this index, in which sequence the elements of the series are connected with each other.

So, for instance, the rule to construct an adic is always starting at index 0; either you add, multiply, even divide, you have to create an operation, which is able to construct the result from the tail.

As I said, to an adic an actual value cannot be assigned, because they are representing infinities. Some may now say: "yes, actual value is infinity" - but that’s not, what I mean. And even with this statement there is won nothing: all infinities are equal - so what about dealing with the adics at all?

The only thing, that we can discuss on constructions, which represent infinities are global operations. They can have other properties, than an actual value. For instance:

...2222222 can be said to be divisible by 2, ....333333 can be said to be divisible by 3 and so on.

This may be an interesting question on any level; but if the digits are not sequential like here, then difficulties occur: the adic ... 285714285714 for instance: can it be said, to be divisible by 7? Analyzing the adic by proceeding from the tail to the head, there is no convergence: always after including 6 digits the divisibility occurs, if the set of 6 digits is nto completed, then the adic seems not to be divisible.

This example does not kill the "divisibility"-check generally; converting an adic (which is generated on the basis of 10) into one of basis 8 then the number ....9999 will be represented like ....77777, where a digit-by-digit check can be done.

But all this is not a very serious suggestion - I did not play around with this type of properties and also did not came across a discussion of that in AP’s postings. It only shall set the focus on the statement, that with infinities it makes no sense to discuss actual values. The only appropriate way to deal with them is to analyze their number-analytical properties. I want to say it on this prominent place, because one could get confused by the following, where I relate specific adics to specific real values. An adic ....9999 does not have the same value like -1 , but it’s properties regardly addition, multiplication etc may be the same like that of -1.

What can be the highest adic? - That one, where all digits are 9: .....99999. Ok, if you said it’s the highest; what if I add 1? All numbers switch to 0 : ....00000. What does it mean? Is ....99999 related to -1?

Another check: set â = ...9999, then take â - 10*â which is .....99999 - ...9990 ° 9

Analytical this means, -9â ° 9 and then follows again â ° -1

I did not check, whether there are different possible relations, at least they are not obvious.

Ok, if â is related to -1 , then how to express -2,-10,-100? Just decrement the adic:

â -1 = ...99998 ° -2 ; â - 9 = ... 99990 °-10, â - 99 = .... 99900 ° -100 ....

â +1 = ... 0000 ° 0; â+2 = ...001 ° 1 etc.

This last line shows, that even it may be not correct, it is at least convenient, to assume â having the properties of -1, and assuming the other shown adics the properties of the related numbers, which are found using the common notation of digits-addition.

Now, since we have, that we try to take -1 as a relative to â, and are going to use ordinary additions and subtractions of naturals to represent other numbers, what does it mean, if â is divided by, let say 3? The digitline looks like ...3333; and the numerical value of -1 divided by 3 is -1/3. We also could try with a division by 9, which makes the result to ...1111 with a relation to -1/9. ...7777 then could be represented by -7/9 etc. These few examples look not to be too much; one would expect, that if there is any sense in it, that at least all representations for the rationals between 0 and -1 should be calculatable: further constructions could be something like ...090909 or ...54321, for which an appropriate representation could not be given in this chapter.

Addition, subtraction: to add or subtract integers to an or from an adic could be allowed. Just without further insight, it looks, as if no problem occurs, if integers are added or subtracted to an adic - always remember: we do as if there was a value for the adic, which is incremented or decremented. But this should not be mixed: without recurring to an actual value, we only try to describe the range of consistent operations on adics. To say for instance, â /3 = ê may suggest the idea to rearrange the formula and come out with â / ê = 3 . We are on divergent series - and infinities divided by infinities are NOT defined; and I don’t stand for the opposite opinion.

Suppose, we have the adics-representation for all rationals between 0 and -1; then just by addition of a natural, we can create representances for all rationals between 0 and any positive natural number. By a simple subtraction of a natural we can also create adic-representations for all remaining negative rationals.

If ....9999 represents -1, then ....9998 represents -2, ...99990 represents -10 and so on.

If ....11111 represents -1/9 then +8/9 can be represented by ....11112; +1/9 again can be created by (-8/9)+1, so we can write ....888889 represents +1/9.

Now addition of two adics. I did not check anything outside the here written for contradictions; but everything seems to be consistent after applying addition, say ...1111 + ...88888 . Formal digit-operations for additions and subtractions did not produce an inconsistency in the here discribed adics-concept.

So even ...8888 + ...11113 comes out to be a valid representations for the actual value, which would be achieved, if the operation would be done with the related values: ...0001 which means 1 and is calculatable by its relatives (-8/9) + (-1/9 + 2).

Multiplication digit by digit from the tail also seems not to conflict with the supposed relation between adics and rationals - even the multiplication of two adics, although the carry from the multiplication when proceeding from the tail to the left increases to infinity as well as the whole term.

Division is more difficult; we are used to an operation from left to right; and with adics the left is not seen. If we have ...9999 / 3 then we can do it on base of each digit; but ...9999 / 7 will not allow such an operation. We have two options: either we can convert the given operand into an adic with another base (for instance 8 with ...77777 as a representation for -1) so that we can do an each-digit operation or we have to find out another algorithm for division, which allows to proceed from the tail.

If we ask: which adic, multiplied by 7 could result in ...99999? and then start by try and error on the tail:

... ?????7* 7 = .... ??????49
....????57* 7 = .... ?????399
....???857* 7 = .... ????5999
....??2857* 7 = .... ???19999
....?42857* 7 = .... ??299999
....142857* 7 = .... ??999999

and now...

....142857 142857 142857* 7 = .... 999999 999999 999999

with ....142857 142857 142857 given the adics-representation for a -1/7-value.

While it seems, that the adics can be related to the negative rationals between 0 and -1, although the notation is without a negative sign, positive rationals can be related just by formal addition of naturals.

Another need for the description of related positive rationals occur, if we do the simple task of squaring an adic; for instance multiplying -1*-1=1 and see, what adic-representation will be related to this term:

If multiplication is performed, then that means to do

...99999 * ....99999 , which one can perform from the rightmost number:

99999 * 999999
---------------------------
99991
99991
99991
......
----------------
..00001,

which we already assumed to be the adic related to 1. If we use

....111111* .... 111111

then the result is something like

... 5432 098765432 0987654321, which is the adic for (-80/81) + 1.

From the given examples it seems, that the adic-representation for negative rationale shows always periodic digit-sequences. I did not test that, but I assume, that this must be. This is, however, always thought respectively an "addition" of naturals, which introduce an offset before the preriodicity begins ( ....111112 with periodicity of 1 and offset of 1, because the period starts after the first digit). The adics-representation for -1/7 is also periodic, with a period as shown above.

If adics and decimals are mixed, then constructions like ...1111.11111... occur. I didn’t see AP discussing this matter in his papers, but I do not read regularly, so maybe I missed something.

Adics with decimals, where the decimals are the same positive representations for rationals as the adic-part is the negative representation, should represent the value 0. As there are many ways to compensate the adic-rational-representation by a decimal, the decimal-adic-construct is not

Next question could be, to determine adics as well for irrational values. I did not try that really, but, for instance sqrt(2/3) could be representated by something sqrt(-1/3+1) or sqrt(...33334). As the division of adics is already an indirect task of trial and error, the square-rooting should be a major task. The numbers could start for example with:

12 * 1 2
144