# What are Santilli's isonumbers?

A friend of mine asked me yesterday about Santilli's isonumbers. I told him that it was quackery. As I based my answer only on the general reputation of the guy and had no knowledge of the subject, I decided to ask this question here.

Question: What is the isonumber? Did any serious mathematician spend any time looking at isomnumbers? What is the conclusion? References would be useful.

I looked at Jiang's monograph for a little while last night. Here is what I could get from it (I am now quoting from memory, so my terminology and notation may not be exactly the same). If $F$ is a field (of "numbers"), then the field $\overline{F}$ of "isodual numbers" has the same underlying set and addition operation, but multiplication is replaced by the operation $x \ \overline{\bullet}\ y := - (xy)$. The new multiplicative identity is $-1$.

This is mathematically valid, of course: i.e., $\overline{F}$ really is a field. Moreover it is isomorphic to $F$ via the map $x \mapsto -x$, although I couldn't find a clear statement of that. (But somewhat later on I saw references to the isotopy $F \rightarrow \overline{F}$.) Physically speaking, the isodual numbers are supposed to bear the same relation to the ordinary numbers as antimatter does to matter. (I don't know what that means, but I am not a physicist and so am not even going to worry about it.)

Jiang defines a new function $J_2(\omega)$, which is supposed to be some sort of repaired version of the Riemann zeta function. In one of his published works, he claims that the Riemann hypothesis is false -- in fact, he says, the zeta function has no zeros in the critical strip. [Logically speaking, wouldn't that make the Riemann Hypothesis true? Never mind.] From this definition, he immediately deduces proofs of Goldbach, twin primes, primes of the form $n^2+1$, and several other outstanding number theoretic conjectures -- literally immediately, in that I could find no argumentation for them. First these results are stated for "isonumbers" but later on they are stated for the usual integers.

That's about as far as I got. I also noticed, though, that many of the results described in this monograph were first published as papers by the journal Algebra, Groups and Geometries (founding editor: R.M. Santilli). These papers appear on MathSciNet but are not (going to be) reviewed.

Looks like total crackpottery.

• Did you actually understand the definition so that you can say what they are in normal language and why they are of no interest? – Bugs Bunny Jul 26 '10 at 5:49
• I read enough of his overview paper to come to that determination. The initial abstract algebra is fine, but it doesn't do what it's claimed to do. – Charles Jul 26 '10 at 17:06
• Did you read Santilli's paper or Jiang's book? The latter is indeed useless but I cannot access the former one for now... – Bugs Bunny Jul 26 '10 at 19:24
• With all due respect, it seems to me that this doesn't answer the question. – Todd Trimble Jan 21 '15 at 17:10
• @ToddTrimble: True enough -- it addresses only the third question, not the first two. But at the time it seemed like there might not be any other answers. – Charles Jan 21 '15 at 18:13

## protected by Todd Trimble♦Jan 21 '15 at 17:07

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