RSA 704 Question

[philmoore]

Originally Posted by R.D. Silverman

What disk "containing the factors"? Where did I EVER write that the *factors* were written to disk?

You clearly did not, and you apparently did not read my post carefully if you thought I was saying that you did. My quote of your previous post was by way of explaining how I had jumped to a false conclusion.

Originally Posted by R.D. Silverman

Please explain how the factors can be easily *verified* as prime?

Well, here are the reported factors of RSA640:
1634733645809253848443133883865090859841783670033092312181110852389333100104508151212118167511579
and
1900871281664822113126851573935413975471896789968515493666638539088027103802104498957191261465571
Primo will easily produce a certification that each of these numbers is prime that can then be validated using another program. Or we could use CYCLOPROV, or factor P-1 or P+1 and use the "classical" methods. My point is that each of these methods will reach the same conclusion, that both of these numbers are prime. In a scientific sense, this is verification, when we do other experiments that support the original claim.

Originally Posted by R.D. Silverman

There is always a small probability of an error in any computer and computer code that tests them. Multiple independent computers reduce the probability, but it still isn't 0.

So I therefore ask: when does the result of a computer computation constitute "fact"? (It was not I who raised this issue)

But it is an interesting issue just the same. I would say that it constitutes fact when it can be independently verified, perhaps through the use of other computer computations. But then again, perhaps this is more a scientific definition of fact than a mathematical one.

[R.D. Silverman]

Originally Posted by philmoore

You clearly did not, and you apparently did not read my post carefully if you thought I was saying that you did. My quote of your previous post was by way of explaining how I had jumped to a false conclusion.

However, even then, the false conclusion that you claimed was misleading.

There is a difference between what you wrote:

"I had falsely assumed that disks containing the factors had been destroyed."

and

"I had falselt assumed that the disks that were destroyed contained the factors".

The former still implies that the "disks contain the factors".

You used, what a grammar teacher would call "misplaced modifiers".

[rdotson]

Originally Posted by retina

I think I got my answer.

LOL. Mathematics is a very precise science, and I suppose that in order to be a good mathematician one must be very attentive to details. I suspect that those who are good at math have little patience with those who don't value precision (in speech as well as in math), and in some cases don't even know the subject matter well enough to form a sensible and accurate question.

I remember that in my very first question on a Usenet math forum, I incorrectly referred to an equation as an expression, and referred to a variable as being "positive" rather than "positive and non-zero." Chastisement for those two sins dominated the entire thread and my original question was lost in the confusion.

If anyone's interested, the thread is located at: http://xrl.us/s67y

Here was my question:

For: y^2 = x^2 + bx + c
The values of b and c are known non-zero integers.
Is it possible to find at least one integer value of x for which y is also an integer?

[Note: One example is: b=876, c=10118146921, and x=5058977100
y^2 = 25593253740106496521 is a perfect square because it has an integer square root of y = 5058977539]

[philmoore]

Originally Posted by R.D. Silverman

The former still implies that the "disks contain the factors".

Not at all! If I had said "I had falsely assumed that the disks containing the factors had been destroyed" then the assumption that such disks had originally existed would have been a logical implication, but as written, nothing in that statement contradicts your account of how the computation was actually done.

[R.D. Silverman]

Originally Posted by rdotson

Here was my question:

For: y^2 = x^2 + bx + c
The values of b and c are known non-zero integers.
Is it possible to find at least one integer value of x for which y is also an integer?

This reduces to a factoring problem. Complete the square, multiply by 4
and one gets: 4y^2 - (2x + b)^2 = 4c - b^2. --> y1^2 - x1^2 = 4c-b^2.
Now factor 4c-b^2.

[rdotson]

Originally Posted by R.D. Silverman

This reduces to a factoring problem.

Ah, I had a strong hunch that would be the case but didn't know how to prove it, and consequently was hoping the solution might be something of lesser difficulty than factoring. Thank you for the clarification Mr. Silverman.

[cheesehead]

Originally Posted by R.D. Silverman

However, even then, the false conclusion that you claimed was misleading.

I would agree that the usage was not the least-easily-misunderstood, but not that it would inevitably mislead careful readers.

There is a difference between what you wrote:

"I had falsely assumed that disks containing the factors had been destroyed."

< snip >

The former still implies that the "disks contain the factors".

A rare slipup, I think, Dr. Silverman.

In the first sentence, what philmoore states was falsely assumed was the entire clause "disks containing the factors had been destroyed". If no disks containing the factors ever existed, then it is equally true that no disks containing the factors either had been or could have been destroyed, because the latter would have required that such disks have existed before their destruction.