[QUOTE=Dubslow;327867]'pi' is the smallest strictly positive solution x of the following equation defined over the reals: [tex]\sum_{i=0}^\infty {\frac{(-1)^i x^{2i+1}}{(2i+1)!}} \quad = \quad 0[/tex] (this happens to be the ratio of the circumference to diameter of a "circle", i.e. the set of points [tex]S^1 = \{(x,y)\in \mathbb{R}^2 | x^2+y^2=r^2,\quad \forall r \in\mathbb{R}\backslash \{0\} \}[/tex]. If you really want to get fancy, the circumference and diameter can be defined by some suitably abstract integral [but I haven't had enough analysis to say much more than that].)

Now, what is your formal specification/explicit definition of a prime larger than M48? Something like Ackerman(M48, M48) is a formally specified number, but I highly doubt its primality (not that you or I could [i]prove[/i] it one way or the other). It's quite a bold claim you made (or so it seems to my as-yet-untrained senses).[/QUOTE]

Let W(n) be the largest prime factor of n. I assume that you accept this
as a well-defined arithmetic function? Its domain and range are both N.

W(M48! + 1) is a explicit definition of a prime larger than M48.

I hope this group does not make the mistake of confusing a number
with its representation. Many numbers (e.g. pi) have an explicit
definition, but not a decimal representation that can be given.
If we only accept that "knowing" a number means knowing one explicit
representation, (in this discussion, [i]decimal[/i]), then we would have
explicit representations for few numbers indeed.

[QUOTE=R.D. Silverman;327852]Yes, I do. I can give an explicit definition.

In formal automata/language theory it can be specified exactly. (or
as you say, "explicitly"). Of course if one wants to be fuzzy in the definition
of the word "explicit" then it sort of becomes meaningless.

It just depends on the (formal) language one is willing to accept.

If one only accepts decimal (or binary), then one is going to be
very limited in the numbers one can specify. i.e. try specifying 'pi'.[/QUOTE]
Apples and oranges ... pi is transcendental, whereas integers are representable via finite digit strings.

So show us such a largest "known" number, in the sense of producing it via explicit digit string. Or is that too formal for you?

[QUOTE=Dubslow;327867]'pi' is the smallest strictly positive solution x of the following equation defined over the reals: [tex]\sum_{i=0}^\infty {\frac{(-1)^i x^{2i+1}}{(2i+1)!}} \quad = \quad 0[/tex] (this happens to be the ratio of the circumference to diameter of a "circle",

<snip>
[/QUOTE]

Bingo! But (as implied by Ernst) this is not an 'explicit' definition,
because by 'explicit' he meant/implied DECIMAL NOTATION, i.e.
accepting only a decimal point and the digits 0....9 in one's alphabet.

In another post I give an explicitly defined prime that is larger than
M48.

[QUOTE=ewmayer;327884]Apples and oranges ... pi is transcendental, whereas integers are representable via finite digit strings.

So show us such a largest "known" number, in the sense of producing it via explicit digit string. Or is that too formal for you?[/QUOTE]

I can not give Ackerman(M48, M48) as an explicit digit string either.
Yet it too is a well defined integer. Do you not accept it as an
explicit definition of a number???

There is no such thing as the largest known integer, just as there is
no such thing as the largest known prime, because as soon as we
give it we immediately know one larger.

You seem only willing to accept that "knowing a number" means
"giving an explicit digit string". This philosophy does not fit in with
modern mathematics.

Knowing a number does NOT mean "knowing a finite representation of
the number as an explicit [i]digit string[/i]"

I never said I required decimal ... binary, base-3, octal, decimal, hex... all are welcome, it's a big tent.

(circumference of circle) / (diameter of circle) ... oh look, I just explicitly specified pi to infinite precision!

(Although I should've specified "circle on nonzero diameter" ... ssh, don't tell Don Blazys).

[QUOTE=R.D. Silverman;327890]You seem only willing to accept that "knowing a number" means
"giving an explicit digit string". This philosophy does not fit in with
modern mathematics.[/QUOTE]
But you know we're not operating in a broad "modern mathematics" context here ... we're talking about primality proving, you are intimately familiar with the "philosophy" of that special subdiscipline, and thus are merely engaging in pompously obtuse trolling to try to get a rise out of folks and/or belittle others' successes.

If you're bored, why don't you go write a paper or do something actually useful?

[QUOTE=R.D. Silverman;327890]Knowing a number does NOT mean "knowing a finite representation of the number as an explicit [i]digit string[/i]"[/QUOTE]

Please forgive us Dr. Silverman. We're clearly much more stupid than you.

On the other hand, we were in the international news today, and you weren't.

Oh, and you don't have a doctorate, while many of us here do....

[QUOTE=chalsall;327898]Please forgive us Dr. Silverman. We're clearly much more stupid than you.

On the other hand, we were in the international news today, and you weren't.

Oh, and you don't have a doctorate, while many of us here do....[/QUOTE]

Come on dude, we're talking math here, not who does or does not have press coverage and/or patient hope for a degree.
____________________________________________________________________

I agree that Ackermann(M48, M48) is a well defined definition of a number: we know exactly what number we mean without ambiguity, even if we can't write it out. However, I would intuitively argue that W(M48! + 1), while being a perfectly well defined arithmetic function, does not mean we know what number it will output.

I'm trying to come up with a good difference between the two, but the best I've come up with is that evaluating A(M48, M48) requires only addition and subtraction, while calculating W(M48!+1) is not so simple (even if it would be less computer ops overall). Edit: Perhaps it's the fact that factoring is, in all known cases, (either non-deterministic [NFS, ECM], or unknown time to completion [Fermat's method has a firm theoretical upper bound, but could also terminate anywhere between 1 and the upper bound]). Addition, is "doubly deterministic" in that we know exactly how long it will take for all operands (i.e., it terminates, but its time to termination is also deterministic).

[QUOTE=ewmayer;327894]I never said I required decimal ... binary, base-3, octal, decimal, hex... all are welcome, it's a big tent.

(circumference of circle) / (diameter of circle) ... oh look, I just explicitly specified pi to infinite precision!
[/QUOTE]

Aha!. You just gave an explicit definition without using radix notation.
So it appears that you do accept non-radix representations of a number
as being 'explicit'.


W(n) = largest prime factor of n.

W(M48! + 1). oh look, I just defined a larger prime than M48 to
infinite precision!


[QUOTE]
(Although I should've specified "circle on nonzero diameter" ... ssh, don't tell Don Blazys).[/QUOTE]


And you should have said "circle embedded in a Euclidean Plane", or
"circle embedded on a Riemann surface with Ricci Tensor = 0 everywhere"
For example, a circle on the surface of a sphere only has circumference/diameter = pi in the limit as its radius goes to 0.

[QUOTE=Dubslow;327899]Come on dude, we're talking math here, not who does or does not have press coverage and/or patient hope for a degree.[/QUOTE]

You don't understand the history with Mr. Silverman.

[QUOTE=chalsall;327901]You don't understand the history with Mr. Silverman.[/QUOTE]

I understand at least some of the history, and I'm willing to extend at least partial amnesty. He has not done anything particularly un-called-for yet. We are discussing (the philosophy of) mathematics.

[QUOTE=R.D. Silverman;327900]"circle embedded on a Riemann surface with Ricci Tensor = 0 everywhere"[/QUOTE]

..."which everyone knows is the contraction over middle indices of the more general rank-4 Riemann-Christoffel curvature tensor...blah, blah, Theorema Egregium ... blah, blah-blah, R[sub]ij[/sub] only captures all the independent components of R[sup]l[/sup][sub]ikj[/sub] in <= 3 dimensions ... blah, blah..."

To quote the Pythons, "Gosh, we're all really [b]impressed[/b] down here, I can tell you."