Improving On Euler's Conjecture

[David Sklansky]

He was wrong when he said that three fourth powers could never add up to a fourth power.

But what about three sixth powers adding up to a sixth power?

This post is related to an idea that has been kicking around in my head for forty years. I made a misguided effort to introduce this idea with a pevious post about Fermat's Theorem. I'll try again using the above two conjectures. Without going into too much detail I will say that my idea separates number theory questions into two categories. The above conjectures are in different categories. As a hint to what I am driving at, if I conjectured that EIGHT sixth powers could never add up to a sixth power it would be a question in the same category of Euler's original and wrong conjecture. (My guess is that this third conjecture is wrong also. But not the second one.)

[wheeler]

Your "idea" is sufficiently vague that it's almost certainly correct. Or incorrect. How can we tell? You have to be more precise. Mathematicians don't usually explain ideas by not going into "too much detail" and only giving "hints" to what they're trying to say.

For example, from what you've written above I might conclude that you want to just divide conjectures into two categories: those that are right and those that are wrong! But that's certainly not what mean.

Couldn't you try to explain what your two categories of conjecture are, so that we can help you massage your meta-conjecture into a form that has an answer?

- wheeler

[David Sklansky]

For now I'd like to not muddy the waters and merely see what mathmeticians have to say about the second conjecture.

[mslif]

I am trying to follow what you are saying (hopefully I am not completely off, if so please be kind):

I do not think that it will be in the same cathegory because your are talking about:
a^6+b^6+c^6+d^6+e^6+f^6+h^6+i^6=j^6
Euler's Conjecture is:
a^4+b^4+c^4=e^4

Euler proposed that for every integer greater than 2, the sum of n – 1 nth powers of positive integers cannot itself be an nth power.

The fact that you are conjuncturing EIGHT sixth powers does not follow Euler's theory.

[wheeler]

[ QUOTE ]
For now I'd like to not muddy the waters and merely see what mathmeticians have to say about the second conjecture.

[/ QUOTE ]

The second conjecture is open.

The third conjecture is false. See
http://mathworld.wolfram.com/DiophantineEquation6thPowers.html
for counterexamples.

[PairTheBoard]

[ QUOTE ]
Your "idea" is sufficiently vague that it's almost certainly correct. Or incorrect. How can we tell? You have to be more precise. Mathematicians don't usually explain ideas by not going into "too much detail" and only giving "hints" to what they're trying to say.

For example, from what you've written above I might conclude that you want to just divide conjectures into two categories: those that are right and those that are wrong! But that's certainly not what mean.

Couldn't you try to explain what your two categories of conjecture are, so that we can help you massage your meta-conjecture into a form that has an answer?

- wheeler

[/ QUOTE ]

From David's Fermat Thread, I believe his metaconjecture goes something like this:

"There exist True Number Theory Statements that are Impossible to prove."

I think this goes beyond Godel who showed that within any logical system rich enough to include the integers there will exist statements that will be impossible to prove or disprove from within that logical system. David asserts there will be such statements that are in fact true, yet impossible to prove.

I'm really not sure what to make of his conjecture myself. I'm a bit bothered by what his use of the term "true" means.

PairTheBoard

[PairTheBoard]

The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[/ QUOTE ]

Also, even if the Sklansky Metaconjecture could be proven and thus the Skansky Class of number theory conjectures shown to be non empty, it would be impossible to ever show that a particular number theory conjecture actually fell in the Sklansky Class.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[/ QUOTE ]

Also, even if the Sklansky Metaconjecture could be proven and thus the Skansky Class of number theory conjectures shown to be non empty, it would be impossible to ever show that a particular number theory conjecture actually fell in the Sklansky Class.

PairTheBoard

[/ QUOTE ]


I suspect where David would want to go from that point is to develop some kind of theory that would produce "probabilities" of Number Theory Conjectures falling in the Sklansky Class.


He might then open a world wide online Casino to take bets on the action. Unfortunately the only bets that could ever be won would be those against a conjecture belonging to the Sklansky Class.

PairTheBoard

[jason1990]

[ QUOTE ]
From David's Fermat Thread, I believe his metaconjecture goes something like this:

"There exist True Number Theory Statements that are Impossible to prove."

I think this goes beyond Godel who showed that within any logical system rich enough to include the integers there will exist statements that will be impossible to prove or disprove from within that logical system. David asserts there will be such statements that are in fact true, yet impossible to prove.

I'm really not sure what to make of his conjecture myself. I'm a bit bothered by what his use of the term "true" means.

PairTheBoard

[/ QUOTE ]
I don't think it goes beyond Godel. Suppose the conjecture

"There do not exist three sixth powers adding up to a sixth power."

is impossible to prove or disprove. Then it is true. Why? Well, suppose it is false. Then there exists three sixth powers adding up to a sixth power. But if they exist, then there exists a finite proof that this conjecture is false. But that cannot be, because we assumed the conjecture was impossible to disprove. Therefore, it is impossible to prove that this conjecture is impossible to prove or disprove, because that would be a proof of the conjecture. But there may be statements of this form which are impossible to prove or disprove, and whatever statements those are, they are true.

From "Goedel's Proof" by Ernest Nagel and James R. Newman: Goedel showed "that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic approach to number theory, for example, cannot exhaust the domain of arithmetical truth."

[David Sklansky]

"Also, even if the Sklansky Metaconjecture could be proven and thus the Skansky Class of number theory conjectures shown to be non empty, it would be impossible to ever show that a particular number theory conjecture actually fell in the Sklansky Class."

PairTheBoard

We are jumping the gun here but you seem to have the gist of where I am going. Would have preferred that the first person to do this did not believe in Santa Claus. But life is not perfect. Anyway it is the non Sklansky Class that is relevant. Sklansky class statements may are may not be, true, yet unprovable. Non Sklansky class statements must be untrue or provable (if my idea is correct). The first (Euler's)and third conjectures fall into this class. I think.

[PLOlover]

[ QUOTE ]
I don't think it goes beyond Godel. Suppose the conjecture

"There do not exist three sixth powers adding up to a sixth power."

is impossible to prove or disprove. Then it is true. Why? Well, suppose it is false. Then there exists three sixth powers adding up to a sixth power. But if they exist, then there exists a finite proof that this conjecture is false. But that cannot be, because we assumed the conjecture was impossible to disprove. Therefore, it is impossible to prove that this conjecture is impossible to prove or disprove, because that would be a proof of the conjecture. But there may be statements of this form which are impossible to prove or disprove, and whatever statements those are, they are true.

From "Goedel's Proof" by Ernest Nagel and James R. Newman: Goedel showed "that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic approach to number theory, for example, cannot exhaust the domain of arithmetical truth."


[/ QUOTE ]

I'm pretty sure though that you can develop a system which would include a certain conjecture, and thus render it provable or not. It's just that you can't have a general axiomatic set that will encompass all conjectures that are provable, etc.

So yes I think the Sklansky thing is different.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
From David's Fermat Thread, I believe his metaconjecture goes something like this:

"There exist True Number Theory Statements that are Impossible to prove."

I think this goes beyond Godel who showed that within any logical system rich enough to include the integers there will exist statements that will be impossible to prove or disprove from within that logical system. David asserts there will be such statements that are in fact true, yet impossible to prove.

I'm really not sure what to make of his conjecture myself. I'm a bit bothered by what his use of the term "true" means.

PairTheBoard

[/ QUOTE ]
I don't think it goes beyond Godel. Suppose the conjecture

"There do not exist three sixth powers adding up to a sixth power."

is impossible to prove or disprove. Then it is true. Why? Well, suppose it is false. Then there exists three sixth powers adding up to a sixth power. But if they exist, then there exists a finite proof that this conjecture is false. But that cannot be, because we assumed the conjecture was impossible to disprove. Therefore, it is impossible to prove that this conjecture is impossible to prove or disprove, because that would be a proof of the conjecture. But there may be statements of this form which are impossible to prove or disprove, and whatever statements those are, they are true.

From "Goedel's Proof" by Ernest Nagel and James R. Newman: Goedel showed "that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic approach to number theory, for example, cannot exhaust the domain of arithmetical truth."

[/ QUOTE ]

Yes. Nicely done. I think you are right.

Interesting, not only would such a statement be impossible to prove or disprove, but it would be impossible to prove that that's the case.

PairTheBoard

[PairTheBoard]

DS --
"Sklansky class statements may are may not be, true, yet unprovable."

When you say "unprovable" here I take it to mean the following:

Sklansky Class statement include those that are True but are impossible to prove so, and those that are False but are impossible to prove so.

It's clear that a statement that is False but unprovably so, belongs in the Skalnsky Class because it's converse must then be True and unprovably so.

DS --
" Anyway it is the non Sklansky Class that is relevant."

Relevant for what?

DS --
" Non Sklansky class statements must be untrue or provable (if my idea is correct). "

Now when you say "provable" I'm not sure what you mean. Do you mean, True and provably so? Are you saying that Non Sklansky Class statements must either be False or (True and provably so)? They may be False and NonSklansky but they must be provably False. If they are False and unprovably so they are Sklansky Class.

Non Sklansky Class statements are simply ones that are either True or False and provably so. You say these are the relevant ones? I don't see how you've taken us very far.


PairTheBoard

[David Sklansky]

An example of a non Sklansky conjecture would be that the digits of pi never contain exactly 43 consecutive sevens. It must be false or provably true. An example of a Sklansky conjecture about pi would be (if I did my math right) that you will never reach a sequence that exactly duplicates all the previous numbers in order. That, I think, can be true without a proof existing. See the difference?

[PairTheBoard]

[ QUOTE ]
DS --
"Sklansky class statements may are may not be, true, yet unprovable."

When you say "unprovable" here I take it to mean the following:

Sklansky Class statement include those that are True but are impossible to prove so, and those that are False but are impossible to prove so.

It's clear that a statement that is False but unprovably so, belongs in the Skalnsky Class because it's converse must then be True and unprovably so.

DS --
" Anyway it is the non Sklansky Class that is relevant."

Relevant for what?

DS --
" Non Sklansky class statements must be untrue or provable (if my idea is correct). "

Now when you say "provable" I'm not sure what you mean. Do you mean, True and provably so? Are you saying that Non Sklansky Class statements must either be False or (True and provably so)? They may be False and NonSklansky but they must be provably False. If they are False and unprovably so they are Sklansky Class.

Non Sklansky Class statements are simply ones that are either True or False and provably so. You say these are the relevant ones? I don't see how you've taken us very far.


PairTheBoard

[/ QUOTE ]

The wrinkle here, as jason1990 pointed out, is if the statement is of the form:

There exist no integers a1,...an satisfying S(a1,...,an,) where S takes a fixed number, n, of arguments. All these Fermat-Euler type conjectures are of this type.

For statements such as this, if they are False they must be provably false. The reason being that if such a1...an exist a computer can just search till it finds them so they can theoretically be proven to exist.

Therefore, as jason1990 pointed out, all such statements that are of Sklansky Class must be True.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
An example of a non Sklansky conjecture would be that the digits of pi never contain exactly 43 consecutive sevens. It must be false or provably true. An example of a Sklansky conjecture about pi would be (if I did my math right) that you will never reach a sequence that exactly duplicates all the previous numbers in order. That, I think, can be true without a proof existing. See the difference?

[/ QUOTE ]

Why is the first example necessarily NonSklansky?

It's of the type I mention above, which means it is either True or Provably False. In other words, if it is Sklansky Class it must be True, as shown by Jason1990.

While the second example does not obviously fit my Fixed Arguments type statement, it still lends itself to Jason1990's logic. If false it must be provably so because if such a sequence exists a computer can theoretically find it if it searches the required string of numbers. So as far as what we Know apriori about it's Sklanskyness it's in the same category as the first example. If it's Sklansky Class it must be true.

Unless the First Example has already been solved, if you can show me why it is Not Sklansky Class I'll be impressed. Funky hand waving psuedo probabilty arguments don't count.

PairTheBoard

[jason1990]

[ QUOTE ]
So yes I think the Sklansky thing is different.

[/ QUOTE ]
Godel showed that there are statements which are impossible to prove or disprove.

PairTheBoard thought David was asserting something different by saying that some such statements are actually true.

But Godel already showed that. So there is nothing different here and it's a little premature to use the term "Sklansky Class Conjecture" instead of "Godel Statement." But perhaps PairTheBoard has misinterpreted David's assertion. After all, David has not come out and said what he means. So maybe there will be something novel here after all. But that can't be inferred from the current content of this thread.

[PLOlover]

[ QUOTE ]
Godel showed that there are statements which are impossible to prove or disprove.

PairTheBoard thought David was asserting something different by saying that some such statements are actually true.


[/ QUOTE ]

It's my understanding that you can't have a complete system where everything is provable. In other words, if you try to have a complete system, some things won't be provable. If you have an incomplete system, many things might be provable, but with slightly different axiomatic sets.

Definitely I could be wrong.

[jason1990]

A complete system is one in which everything is provable.

If you have a consistent logical systetm which is "large enough," then it will be incomplete. If you try to complete it by adding more axioms, you will generate new unprovable statements.

If the system is inconsistent, then everything is provable. This is, of course, a bad thing. (Every statement can be proven true and also proven false.)

[David Sklansky]

I hadn't planned to define my idea this soon but had hoped only to give examples to illustrate. But I see that things are going astray.

The general principle is that a monkey is not destined to type Shakespeare, even with an unlimited number of attempts if each attempt is (sufficiently) less and less likely to succeed. That is an indisputable fact. It is similar to the fact that if you start a three dimensional random walk it is not at all certain that you will EVER come back to the origin (as you would in a two dimensional random walk.) The Gambler's Ruin problem is similar also. You have a finite probability only, of going broke, even playing an infinitely long game, as long as you have an edge.

I believe there is a connection between the above and certain types of number theory problems. For instance the conjecture that no three sixth powers add up to a sixth power. Probability is not the best word to describe the syndrome though. A better word might be "sparseness". Each time you check a higher sixth power to see if it is the sum of three smaller ones it becomes less and less "likely". And the sum of all those "probabilities" converge. Not so when speaking about three fourth powers however, (or four fifth powers etc.) In those cases the "probabilities" decrease, but slowly, akin to the divergent harmonic series.

In the two pi examples I gave previously , the "probabilities" remain steady in the first case, as in the original monkey problem and decrease by a factor of ten each time in the second problem.

The above is a highly unrigorous explanation. I could make it better but not to the satisfaction of a mathmetician. That's why over the years I have brought up the subject with the likes of Martin Gardner, Douglas Hofstadter, the late Ernest Nagel and a few less well knowns. They have taken an interest but didn't bring it to the next step. Meanwhile a mathmetician named Gregory Chaitin has recently worked on similar stuff and will probably keep me from getting any credit if it turns out I have something here.

For a number theory type conjecture to be in what posters are calling the "Sklansky Class" it is necessary that mere "sparseness" can be the "reason" for its truth. But Sklansky class statements can also be false. They can also be true FOR a "reason". (Example: No two fifth powers add up to another fifth power. That is a Sklansky class statement that was eligible to be true merely because of "sparseness" but is in fact true for logical reasons)

A non Sklansky Class statement can't be true because of sparseness (or more rigorously the probability it is true merely because of sparseness is infinitesimal.) If it is true, there has got to be a "logical reason" for it. I think that is the same thing as saying there is a proof. Euler's Conjecture is an example.

I do not claim that it is always clear whether a statement is Sklansky Class or not, or that a procedure exists to always identify if it is. I believe the pi examples are obvious, so I use them.

I also do not know how my sparseness idea exactly jibes with the conclusions of Godel. I do know though that I needed him to show that some things are unprovable for my ideas to have any chance of being right. Before him the thought that mere sparseness could by itself be a reason for certain things to be true about an infinite class of numbers was inconceivaable.

One last way of explaining things. Take an infinite string of random numbers in order. Apply my second conjecture. That you will never come to a substring that exactly duplicates all the numbers before it in order. Obviously the chances I am right is a bit better than eight in nine. Since we are basically adding 1/10 + 1/100 +1/1000 etc. to get the probability that I am wrong.

If the axioms of math do not "connect" pi with my statement, it is still probably true simply because it is true of eight ninths of all infinite strings of numbers. If instead I had said that there will never be a hundred consecutive sixes in pi it would be different, since only an infintesimal fraction of all strings of numbers have that characteristic. So the definition of pi must have caused it to be true (if it was true, which I don't think it is.)

Hopefully there is someone out there who can make these thoughts more rigorous or who will show it to somebody who can.

[Siegmund]

Sparseness is quite a common concept in mathematics, both in number theory and elsewhere.

Pick any property a number can have that you like. It is possible that:

1) No number has this property
2) Finitely many numbers have this property
3) The set of numbers having this property is sparse
4) The set of numbers having this property is infinite in such a way as to not be sparse

and then with 4) you can break it down as to whether the set of numbers NOT having this category is large, sparse, finite, or zero.

Progress is often made toward an impossibility proof by first proving 3) or 2), and the leap from 3) to 2) is often harder than the final impossibility proof.

Proving 3) is enough for many applications: a function that is bounded everywhere but continuous only almost everywhere is integrable; in statistics we often can ignore the possibility of events with probability measure 0 ever happening.

So, yes, you are on to something interesting, in the sense that proving something is sparse can be a significant step forward (though proving that the set of numbers that can be represented by the sum of k-3 kth powers is sparse is a very small step forward, compared to showing something about how many kth powers appear in that set.)

But it is going to be pretty rare to find a problem where sparseness of solutions alone is enough to settle the truth or falsehood of a "there exists..." question.

[Darryl_P]

[ QUOTE ]
If instead I had said that there will never be a hundred consecutive sixes in pi it would be different, since only an infintesimal fraction of all strings of numbers have that characteristic.

[/ QUOTE ]

Are you saying it's unlikely that there are a hundred consecutive sixes in the infinite expansion of pi?

I would lay 1-100 that there is. In a flinch I might add. Reason: ((10^100-1)/(10^100))^n tends to zero as n tends to infinity. I figure you're aware of that as well so maybe I'm misunderstanding you here?

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
If instead I had said that there will never be a hundred consecutive sixes in pi it would be different, since only an infintesimal fraction of all strings of numbers have that characteristic.

[/ QUOTE ]

Are you saying it's unlikely that there are a hundred consecutive sixes in the infinite expansion of pi?

I would lay 1-100 that there is. In a flinch I might add. Reason: ((10^100-1)/(10^100))^n tends to zero as n tends to infinity. I figure you're aware of that as well so maybe I'm misunderstanding you here?

[/ QUOTE ]

I think he's saying just the opposite. If the numbers in pi were random the sequence of sixes would appear with probabilty 1 (infinitely often in fact). Therefore, if it happened that the converse is true for pi, ie. no such sequence of sixes appear, it would be so "improbable" that it would have to be due to something special about pi that could be logically deduced from pi's properties.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
I hadn't planned to define my idea this soon but had hoped only to give examples to illustrate. But I see that things are going astray.

The general principle is that a monkey is not destined to type Shakespeare, even with an unlimited number of attempts if each attempt is (sufficiently) less and less likely to succeed. That is an indisputable fact. It is similar to the fact that if you start a three dimensional random walk it is not at all certain that you will EVER come back to the origin (as you would in a two dimensional random walk.) The Gambler's Ruin problem is similar also. You have a finite probability only, of going broke, even playing an infinitely long game, as long as you have an edge.

I believe there is a connection between the above and certain types of number theory problems. For instance the conjecture that no three sixth powers add up to a sixth power. Probability is not the best word to describe the syndrome though. A better word might be "sparseness". Each time you check a higher sixth power to see if it is the sum of three smaller ones it becomes less and less "likely". And the sum of all those "probabilities" converge. Not so when speaking about three fourth powers however, (or four fifth powers etc.) In those cases the "probabilities" decrease, but slowly, akin to the divergent harmonic series.

In the two pi examples I gave previously , the "probabilities" remain steady in the first case, as in the original monkey problem and decrease by a factor of ten each time in the second problem.

The above is a highly unrigorous explanation. I could make it better but not to the satisfaction of a mathmetician. That's why over the years I have brought up the subject with the likes of Martin Gardner, Douglas Hofstadter, the late Ernest Nagel and a few less well knowns. They have taken an interest but didn't bring it to the next step. Meanwhile a mathmetician named Gregory Chaitin has recently worked on similar stuff and will probably keep me from getting any credit if it turns out I have something here.

For a number theory type conjecture to be in what posters are calling the "Sklansky Class" it is necessary that mere "sparseness" can be the "reason" for its truth. But Sklansky class statements can also be false. They can also be true FOR a "reason". (Example: No two fifth powers add up to another fifth power. That is a Sklansky class statement that was eligible to be true merely because of "sparseness" but is in fact true for logical reasons)

A non Sklansky Class statement can't be true because of sparseness (or more rigorously the probability it is true merely because of sparseness is infinitesimal.) If it is true, there has got to be a "logical reason" for it. I think that is the same thing as saying there is a proof. Euler's Conjecture is an example.

I do not claim that it is always clear whether a statement is Sklansky Class or not, or that a procedure exists to always identify if it is. I believe the pi examples are obvious, so I use them.

I also do not know how my sparseness idea exactly jibes with the conclusions of Godel. I do know though that I needed him to show that some things are unprovable for my ideas to have any chance of being right. Before him the thought that mere sparseness could by itself be a reason for certain things to be true about an infinite class of numbers was inconceivaable.

One last way of explaining things. Take an infinite string of random numbers in order. Apply my second conjecture. That you will never come to a substring that exactly duplicates all the numbers before it in order. Obviously the chances I am right is a bit better than eight in nine. Since we are basically adding 1/10 + 1/100 +1/1000 etc. to get the probability that I am wrong.

If the axioms of math do not "connect" pi with my statement, it is still probably true simply because it is true of eight ninths of all infinite strings of numbers. If instead I had said that there will never be a hundred consecutive sixes in pi it would be different, since only an infintesimal fraction of all strings of numbers have that characteristic. So the definition of pi must have caused it to be true (if it was true, which I don't think it is.)

Hopefully there is someone out there who can make these thoughts more rigorous or who will show it to somebody who can.

[/ QUOTE ]

ok. I think I see what you're getting at now. I think you should read up on how the concept of "sparseness" is already used in number theory according to their definition, and I think you should read everything Chaitin publishes along these lines, then clarify, put your idea down in a rigourous way, and make some well defined conjectures employing the idea. Get one little result of your own, publish it along with the conjectures, get there ahead of Chaitin, and make a name for yourself.



PairTheBoard

[PLOlover]

[ QUOTE ]
A complete system is one in which everything is provable.

If you have a consistent logical systetm which is "large enough," then it will be incomplete. If you try to complete it by adding more axioms, you will generate new unprovable statements.

If the system is inconsistent, then everything is provable. This is, of course, a bad thing. (Every statement can be proven true and also proven false.)

[/ QUOTE ]

Yes, sometimes a triangle has 180 degrees, sometimes <180, sometimes <180 , depending.

But I don't know of a system where the degrees of a triangle are unknowable.

For example.

[Darryl_P]

[ QUOTE ]
I think he's saying just the opposite.

[/ QUOTE ]

I considered this possibility as well, but then in the statement of his I quoted it would make more sense to highlight the finite length and constant, nonzero probability of the sequence rather than the infintesimality of it.

I figured he was a better writer than mathematician so I assumed a math error was more likely. You could be right, though.