Wikipedia:Reference desk/Archives/Mathematics/2010 October 28

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October 28

Obscure 3D shape that folds

In a kids mathematics magazine I used to get back in the early 2000's, there was a free gift of a card net of an obscure geometric shape to fold 'n' glue yourself

the shape had a really long name which i think ended in "hedron" (narrows it down! :-\) which i cannot remember, the magazine may even have been forced to make up the name themselves but it sounded authentic (inside the magazine the name was explained - which brings to mind lots of uses of "hex" and "dec", but my knowledge of shapes since may have distorted this memory.

To add to the challenge I cannot remember the name of the magazine itself, but it had a sister publication of "Young Scientist" (try googling for that lol - its not the first one), which had a version for older children called "SciTec" or "SciTech", and there was an equivalent older kids version of the maths magazine, whose name also escapes me. the publisher of all four of these was Educational publishing, under the name of which i can find at least three different publishing houses, and the relevant one seems to have gone under, since it's website domain is up for sale the editor of the young scientist magazine (at least) was Jenny Smith

every issue of each of the mags came with some kind of science or maths toy, such as 3d glasses. I want to find the name and net of the shape so i can make another one - it folded much like a Paper fortune teller, but on both sides, and looked to be made of cubes with triangular "hinges"

I have looked for the shape on wikipedia - i wouldn't be surprised if there was at least a stub for it, but I am at a loss for how to search with no clue of the name I have googled several combinations of the magazine names to find a publishers website - but as I said it has been taken down replaced with a for sale http://www.educationalpublishing.com/ (.co.uk has a similar notics under a different agency I'm not sure which is correct)

The end goal is finding the shape, not the magazine, but if anyone gets as far as the magazine I'd be interested anyway ☭ мдснєтє тдлк^ЅТЦФФ 00:11, 28 October 2010 (UTC)

Hexaflexagon? -- 119.31.121.84 (talk) 00:45, 28 October 2010 (UTC)

Study our article polyhedron. Bo Jacoby (talk) 09:27, 28 October 2010 (UTC).

This site has something that sounds similar, which it calls a "kaleidocycle" or "flexahedron". the wub "?!" 12:08, 28 October 2010 (UTC)

thanks 119.31.121.84 ! that was it the name seems so obvious now. It was actually Hexahexaflexagon but it was on the Flexagon page you linked. ☭ мдснєтє тдлк^ЅТЦФФ 04:25, 30 October 2010 (UTC)

Another Logic query - uncountability

Hi all, me again! I asked a question on axiomatic deductions a few days ago, but I was curious about something I couldn't figure out on my own. Won't need as thorough an answer this time, I don't actually -need- to do a question on this, I'm just curious :)

In the lecture course I attend we always assume (for propositional logic) that our primitive propositions are countable, and therefore so are all possible propositions. However, I'm asked here what happens when we allow uncountability, and I haven't really got a clue!

We call a set of propositions S independent if for all s in S, we can not deduce s from S-{s} (i.e. s is required to deduce s from S). We call 2 proposition sets S, T equivalent if for every s in S, we can deduce s from T, and for every t in T, we can deduce t from S. I've shown that for every proposition set (countable), there exists an independent set equivalent to it, but now what happens if we allow our initial propositions to be uncountable? Is there still always an independent equivalent subset?

By the nature of the question, I expect there to be a counterexample rather than a proof. Is there any way we can map this into the reals or [0,1] perhaps, then use a more well-known set of results to derive the result? I know you can consider a topological space on the valuations as a Stone space (though I don't know much about stone spaces), hence the name of the compactness theorem, as we reach a conclusion on a compact T.S. However, is there anything we can do in this case, any map into R we can use to help us for example?

Thankyou (time and time again!), 62.3.246.201 (talk) 00:39, 28 October 2010 (UTC)

I'm not terribly well versed in logic, or set theory for that matter, but I believe the uncountable case follows from Zorn's lemma. Take P as the set of all subsets of S, an uncountable set of propositions. Take E as the set of all elements of P equivalent to S, which is clearly nonempty, since it contains s. Partially order E by reverse inclusion. Zorn's lemma provides us with at least one maximal element M, which, from our reverse ordering, is actually minimal under standard inclusion. Any subset M' in E of M would be comparable to M under our partial order, so, since M is minimal, M' must be M. That is, no proper subset of M is equivalent to S, therefore, to M. Take M-{m} for any m, which is now not equivalent to M, so m cannot be deduced from M-{m}. Thus M is independent and equivalent to S.

I wouldn't be at all surprised if this question is (in a more technical sense than I've used) equivalent to Zorn's Lemma, which is itself equivalent to the Axiom of Choice and a number of other things, under the proper set theory. 67.158.43.41 (talk) 04:05, 28 October 2010 (UTC)

Zorn's lemma fails here: the relevant poset is not chain-complete. This can be seen even in the countable case: take S={p₁,p₁∧p₂,p₁∧p₂∧p₃,...}, and let E_n be S with the first n elements removed. Then each E_n is equivalent to S, and they form a chain in E, but the only set contained in all of them is the empty set, which is not in E. Algebraist 09:56, 28 October 2010 (UTC)

Ach, this is so stupid of me. When I stated the uncountable case, I said something different to the countable case! Sorry. I meant to say, in the uncountable case, is there always an equivalent independent set, not necessarily subset? I actually used that very example to show there isn't always such a subset, so this was doubly foolish of me. Does there always have to be an independent set equivalent to the original by Zorn's Lemma then? Thankyou again, 131.111.185.74 (talk) 17:48, 28 October 2010 (UTC)

That's what I assumed you meant, since it's the natural question and the one Imre set. If Zorn's lemma solves the problem, it does so in some rather subtle way; all the obvious approaches I've tried fail. Algebraist 18:23, 28 October 2010 (UTC)

Who's Imre?—Emil J. 10:30, 29 October 2010 (UTC)

Imre Leader, who is fond of this problem and whose undergraduate logic course the OP appears to be taking. Algebraist 10:43, 29 October 2010 (UTC)

Bah, how embarassing. Forgetting to check for bounded chains when applying Zorn's lemma is a bit like making an omelet while forgetting the eggs.... Modifying the above, I've gotten a bit farther, at least. If you take P to be the set of all independent sets provable from S, P contains the singletons from S so is nonempty. Any chain C = {I_a} has upper bound J = Union of C. To the contrary, if J isn't independent, some j in J is proven by J-{j}. Assuming proofs are of finite length, they can apply at most finitely many axioms in J-{j}, so say the finite subset J_0 of J-{j} proves j. J_0 + {j} are each contained in some finite set of I_a's, where the union of this finite set must itself be an I_a since they form a totally ordered finite sequence through inclusion. Call this union I_0. But then I_0 is independent and contains J_0 + {j}, so I_0 - {j} doesn't prove j, so J_0 doesn't prove j, a contradiction. So, each chain is bounded and Zorn's lemma gives a maximal independent set containing elements provable from S. Unfortunately, I haven't been able to show such a maximal set is equivalent to S, or to provide a counterexample. 67.158.43.41 (talk) 02:39, 30 October 2010 (UTC)

I think Martin's Axiom can extend the countable case to sizes less than continuum. It seems like it should fail at size continuum, but I'm having trouble showing that. My first thought: let your language contain a unary predicate $U_{r}$ for every real $r$, and have the axioms $U_{q}\subset U_{r}$ for every $q<r$. This doesn't seem to have an independent axiomatization, but I can't show that.--203.97.79.114 (talk) 20:39, 28 October 2010 (UTC)

Note that the OP asked about propositional logic, so you can't use any predicates. It probably does not make much of a difference, though.—Emil J. 10:30, 29 October 2010 (UTC)

Whoops. In that case, the natural analog would be what you have below.--203.97.79.114 (talk) 10:42, 29 October 2010 (UTC)

This is a nontrivial question. I was actually thinking about the problem some time ago, but IIRC I did not reach any conclusion. It seems that it should not hold in general for uncountable sets of formulas. The specific counterexample I had in mind is to take an uncountable linear order L and put $T=\{p_{u}\to p_{v}\mid u,v\in L,u<v\}$ using propositional variables $\{p_{u}\mid u\in L\}$. However, one can actually show that this has an independent axiomatization for quite a large class of linear orders, so if it works at all, extra conditions are needed. I think that it should be enough to make L an $\aleph _{1}$-saturated dense order.—Emil J. 10:30, 29 October 2010 (UTC)

Does that have an independent axiomatization for L=R? Algebraist 10:44, 29 October 2010 (UTC)

That I don't know. The class I mentioned consists of total orders L which contain |L| (nondegenerate) countable subintervals. One can check that this includes all scattered orders, and more generally, scattered sums of countable orders. In contrast, all intervals in the reals are uncountable, hence it is perfectly possible that the corresponding theory has no independent axiomatization, but on the other hand, the reals are generally quite well-behaved, so I would be rather cautious with making this a conjecture.—Emil J. 12:06, 29 October 2010 (UTC)

Incidentally you're quite right about Imre, I am currently taking his course, he is a fantastic lecturer - unfortunately he always puts an 'extra hard' question at the end of the sheet, II didn't have time to go through it in my supervision on it on this work, I'm just doing it now because I'm a masochist ;-) I'm told the next worksheet's final question took a group of around 10 Cambridge professors a number of hours to solve, so I always have that to look forward to! Now I'm going to try and read through everything you've all written and get my head around it, thankyou everyone for the stimulating discussion thus far! 131.111.16.20 (talk) 13:33, 29 October 2010 (UTC)

It turns out that linear (or partial, for that matter) orders do not work, the theory T above always has an independent axiomatization. Enumerate non-maximal elements of L as $\{a_{\alpha }\mid \alpha <\kappa \}$. For each of them pick an element f(a) > a, and define $\varphi _{\alpha }=p_{a_{\alpha }}\to p_{f(a_{\alpha })}$. Enumerate all pairs of u < v as $\{\langle u_{\alpha },v_{\alpha }\rangle \mid \alpha <\kappa \}$, and put $\psi _{\alpha }=p_{u_{\alpha }}\land p_{f(u_{\alpha })}\to p_{v_{\alpha }}$, $S=\{\varphi _{\alpha }\land \psi _{\alpha }\mid \alpha <\kappa \}$. It's obvious that S ≡ T. For any α, the assignment

$e_{\alpha }(p_{u})={\begin{cases}1&{\text{if }}u=a_{\alpha }{\text{ or }}u>b_{\alpha },\\0&{\text{otherwise}}\end{cases}}$

satisfies φ_β for β ≠ α, and all ψ_β, but it does not satisfy φ_α. Thus S is an independent axiomatization of T.

This can be generalized to an arbitrary theory T as follows. Let κ be the smallest cardinality of a theory equivalent to T (in other words, the number of propositional variables T essentially depends on). Then the following are equivalent:

• T has an independent axiomatization.
• There exists a sequence $\{e_{\alpha }\mid \alpha <\kappa \}$ of assignments such that $e_{\alpha }\nvDash T$, and for every $\varphi \in T$, the set $\{\alpha <\kappa \mid e_{\alpha }\nvDash \varphi \}$ is finite.

(Pretty much the same also holds for first-order logic, with models in place of assignments.) The top-to-bottom implication is easy (take an independent axiomatization, and assignments witnessing its independence). In the opposite direction, we first find formulas φ_α provable in T such that $e_{\beta }\vDash \varphi _{\alpha }\iff \beta \neq \alpha$. This can be done as follows: we pick a formula $\xi \in T$ such that $e_{\alpha }\nvDash \xi$. Let $\{\beta \neq \alpha \mid e_{\beta }\nvDash \xi \}=\{\beta _{1},\dots ,\beta _{n}\}$, and for each i, fix a formula $\zeta _{i}$ such that $e_{\beta _{i}}\vDash \zeta _{i}$ and $e_{\alpha }\nvDash \zeta _{i}$. Then $\varphi _{\alpha }=\xi \lor \zeta _{1}\lor \dots \lor \zeta _{n}$ works.

Enumerate T (or any theory of cardinality κ equivalent to T) as $\{\chi _{\alpha }\mid \alpha <\kappa \}$, and define

$\psi _{\alpha }=\bigwedge _{e_{\beta }\nvDash \chi _{\alpha }}\varphi _{\beta }\to \chi _{\alpha },$

and $S=\{\varphi _{\alpha }\land \psi _{\alpha }\mid \alpha <\kappa \}$. Then it is easy to see that S is equivalent to T, and for any α, the assignment e_α satisfies all φ_β for β ≠ α, and all ψ_β, but it does not satisfy φ_α. Thus S is an independent axiomatization of T.

This also implies that the problem has a topological interpretation after all. Namely, the following are equivalent:

• Every propositional theory has an independent axiomatization.
• For every infinite cardinal κ and every open subset $U\subseteq 2^{\kappa }$, if U is not a union of less than κ compact subsets, then it contains a subset of cardinality κ that has no limit point in U.

Hopefully, some topologist can tell us whether this is true or not.—Emil J. 11:19, 1 November 2010 (UTC)

I don't see why your topological reformulation is equivalent, but I can construct a counterexample to it. By Konig's theorem, cf$(2^{\kappa })>\kappa$. Since compact sets are bounded, $U=2^{\kappa }$ is not a union of even $\kappa$ many compact subsets. However, again by cofinality, every subset of cardinality $\kappa$ has a limit point in $U$. Can you translate this back into a theory with no independent axiomatization?--203.97.79.114 (talk) 20:24, 1 November 2010 (UTC)

That's weird. Why do my kappas look different?--203.97.79.114 (talk) 20:25, 1 November 2010 (UTC)

Wait, you're probably giving $2^{\kappa }$ the Tychonoff topology. In that case, never mind. I was using the Von Neumann cardinal assignment.--203.97.79.114 (talk) 20:54, 1 November 2010 (UTC)

Yes, I mean the product topology. And I'm stupid not to have realized immediately that the property is trivial to prove, even in the original logic formulation. Namely, κ was defined so that T depends on κ variables, let's call them $\{p_{\alpha }\mid \alpha <\kappa \}$. This means that there exist assignments e_α and e'_α that differ only on the variable p_α such that $e_{\alpha }\nvDash T$ and $e'_{\alpha }\vDash T$. Then $\{e_{\alpha }\mid \alpha <\kappa \}$ have the required property, since any φ provable in T can be falsified only by those e_α where p_α appears in φ, and those are finitely many. Thus, every propositional theory has an independent axiomatization.

I think that the same argument also gives independent axiomatizability of all first-order theories. We just replace assignments with models, and propositional variables with symbols in the language. The only slightly non-obvious property is that if T depends on at most κ symbols, then it can be axiomatized by κ formulas. In fact, if L is the set of symbols T depends on, then T is axiomatized by $T\upharpoonright L=\{\varphi \mid \varphi {\text{ is an }}L{\text{-sentence, }}T\vdash \varphi \}$. This follows from Craig's interpolation theorem, but it can also be easily proved directly.—Emil J. 11:10, 2 November 2010 (UTC)

In case the translation is not obvious, here is a direct proof of the topological formulation. Let I be the set of all α < κ such that there exist points x^α ∈ U, y^α ∉ U, such that x^α and y^α only differ on the αth coordinate. If λ := |I| < κ, we can write U = V × 2^κ−I, where V is an open subset of 2^I. Since 2^λ has a basis of cardinality λ, we can write V (and therefore U) as a union of at most λ basic clopen sets, contradicting the assumption. Thus |I| = κ, and I claim that $X=\{x^{\alpha }\mid \alpha \in I\}$ has no limit points in U. If u ∈ U, we can find a basic clopen set $B=\{w\mid w_{\beta _{1}}=a_{1}\land \dots \land w_{\beta _{k}}=a_{k}\}$ such that u ∈ B ⊆ U. Then x^α ∈ B can only hold for $\alpha \in \{\beta _{1},\dots ,\beta _{k}\}$, since $y^{\alpha }\notin B$.—Emil J. 12:23, 2 November 2010 (UTC)

Integral calculus

Given ${\frac {d\sec x}{dx}}=\sec x\tan x$ and $\sec x={\frac {\sec ^{2}x+\sec x\tan x}{\sec x+\tan x}}$, find $\int \sec xdx$. What is the link between the first two identities and the question ? ?--115.178.29.142 (talk) 01:05, 28 October 2010 (UTC)

Hint: How is the numerator of the second identity related to its denominator? More hint; you should try yourself before reading past this word: It's the derivative. 67.158.43.41 (talk) 03:15, 28 October 2010 (UTC)

Unbelievable that I didn't notice that myself. Thanks. 115.178.29.142 (talk) 03:34, 28 October 2010 (UTC)

Tough probability question

If four cards are chosen at random from a standard deck (with all jokers removed, and face cards assigned value as follows: Jack = 11, Queen = 12, King = 13), what is the probability that using the elementary functions of addition, subtraction, multiplication, and division, they cannot be made to equal 24. In other words, for four cards with values a, b, c, and d (not necessarily distinct), what is the probability that using no elementary functions nor arrangements that a ₪ b ♥ c ♯ d (where ₪, ♥, and ♯ are elementary functions, also not necessarily distinct) will be 24. You cannot use the same card more than once, for example you can't say a ♦ a ₪ b ♥ c ♯ d, and you must use all the cards (so a ₪ b ♥ c does not work either). Thanks. 24.92.78.167 (talk) 02:26, 28 October 2010 (UTC)

How are parentheses handled? Aces are 1, I hope? A standard deck has 4 copies of each card, so there are 13^4 = 28561 possible distinct ordered four-card hands. There are four functions you listed. If parentheses are ignored and evaluation is assumed to be (((a ₪ b) ♥ c) ♯ d) there are only 4^3 = 64 choices for each (₪, ♥, ♯), leading to just 64*28561 = 1827904 possible combinations. If parentheses can be supplied arbitrarily, there are a few more possibilities:

((a ₪ b) ♥ c) ♯ d

(a ₪ b) ♥ (c ♯ d)

(a ₪ (b ♥ c)) ♯ d

a ₪ ((b ♥ c) ♯ d)

a ₪ (b ♥ (c ♯ d))

leading to 28561 * 64 * 5 = 9139520 possibilities to compute. These numbers are quite small for computers, so you could write a script to simply compute the answer. I'd be delighted to hear an analytic solution, but I suspect it's a horrifically complicated case-based analysis (or nearly equivalent to what I just wrote). There are a few optimizations. First, multiplication and addition are associative, so many of the extra parenthesizations are actually equivalent. Similarly multiplication and addition are commutative, so many hand rearrangements are equivalent. You could actually cut the run time by an ~eighth by running those (₪, ♥, ♯) made of only multiplication and addition on the set of unordered distinct four-card hands, which is of size (13 multichoose 4) = (13+4-1 choose 4) = 1820. Some combinations can be eliminated by inspection as well; for instance, 4 aces can't possibly get large enough with the operations you've given, and neither can anything with all 2's or under. You also can stop evaluation of a given (a, b, c, d) when you've found a combination that totals 24. I doubt these optimizations are worthwhile in your case, though, considering a script should be able to brute force the problem in a few seconds. 67.158.43.41 (talk) 03:43, 28 October 2010 (UTC)

If anyone else is curious, the OP's question is related to the popular mathematical game 24 Game in which you try to reach 24 given 4 numbers and the permissible mathematical operations above. Zunaid 16:22, 29 October 2010 (UTC)

Pairs of antihomologous points lie on a circle

Could someone please review this section of the Homothetic center article? I believe it to be wrong and there is a discussion on the article's talk page, but would like an expert opinion. If confirmed incorrect, is anything from that section salvageable? Thanks. -- Tom N (tcncv) talk/contrib 04:30, 28 October 2010 (UTC)

Rather wrong proof; correct end result, I think. JoergenB (talk) 21:36, 29 October 2010 (UTC)

Fox calculus on [F', F']

I'm looking at words w in the free group, F, of rank n, with basis x_i, whose images in the abelianisation of F := F^o are trivial. I've modified the domains of the Fox calculus derivatives to be ZF^o, rather than ZF, and I'm trying to show that the only such words w which have zero (modified) Fox derivative with respect to all the basis elements of F are the members of [F ',F '], i.e. commutators of commutators. Unfortunately, I'm not having much luck.

I've tried working in the simple case where we can write $w=ux_{i}u'x_{i}^{-1}u''$, where w only contains x_i (and hence x_i^-1) once. Then the derivative of w with respect to x_i is $u-uu'$ which equals zero if and only if the image of $u'$ is trivial in the abelianisation, which forces the image of $uu''$ to also be trivial. I'm tempted to try some sort of induction argument, but even then, I'm not sure how to get a foothold in the general case, since the Fox derivative is so heavily dependent upon where the x_i 's appear in the word. Any help would be great. (I've considered that the statement I'm trying to prove isn't actually true, but I'm using it to try to flesh out the proof from a textbook, and I believe it boils down to the above situation). Icthyos (talk) 11:40, 28 October 2010 (UTC)

Differential equation

Hi, can anyone see whether there is any closed-form solution to the following?

d^2y/dx^2 = k*y*sqrt(1 + (dy/dx)^2)

k is an arbitrary constant. —Preceding unsigned comment added by 86.184.26.179 (talk) 13:50, 28 October 2010 (UTC)

Just formatting for legibility: ${\frac {d^{2}y}{dx^{2}}}=ky{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}$ —msh210℠ 17:02, 28 October 2010 (UTC)

It's a long time since I did this sort of thing, but I remember being taught to use the substitution $P={\frac {dy}{dx}}$, which gives ${\frac {d^{2}y}{dx^{2}}}=P{\frac {dP}{dy}}$, after which you have a differential equation in P and y only, and can separate the variables. AndrewWTaylor (talk) 17:11, 28 October 2010 (UTC)

Thank you. I end up with an integral that doesn't seem to be solvable in terms of "ordinary" functions, but I guess that's as good as it gets... —Preceding unsigned comment added by 86.173.172.30 (talk) 19:13, 28 October 2010 (UTC)

Maybe try making some trig substitutions ... If I have it right, you now have $\int {P \over {\sqrt {(1+P^{2})}}}dP=\int kydy$, and putting $P=\sin \theta$ gives ${\sqrt {(1+P^{2})}}=\cos \theta$ ... Like Andrew, though, I'm very rusty on this stuff. --81.153.109.200 (talk) 20:02, 28 October 2010 (UTC)

$P=\sin \theta$ certainly does not give ${\sqrt {(1+P^{2})}}=\cos \theta$. The identity you are thinking of is ${\sqrt {1-\sin ^{2}\theta }}=\cos \theta$

Dammit, you're completely right. I was thinking of cosh and sinh ... --81.153.109.200 (talk) 20:46, 28 October 2010 (UTC)

Hint - the integral with P should be easy, even without a substitution. AndrewWTaylor (talk) 22:35, 28 October 2010 (UTC)

Sure, that one's easy; it's at the next stage that things get sticky. 86.173.172.30 (talk) 23:00, 28 October 2010 (UTC)

Allow me to introduce you to my new best friend, who unfortunately this time has come up with a solution that I can barely make sense of. On the other hand copying and pasting your question into Google exactly as written led me to this page. Answer in third post, however some initial conditions are assumed so it's not the most general solution. Zunaid 16:17, 29 October 2010 (UTC)

The equation in the Matlab forum is not the same as here. The y is missing, making it the well-known equation for a hanging chain.

For the equation in the OP, Mathematica gave an answer using the inverse of an elliptic function. But I don't know if that's reliable since it doesn't give any answer when I add initial conditions. -- Meni Rosenfeld (talk) 09:43, 31 October 2010 (UTC)

property of primes

if you picked a random prime of a certain number of large digits, say 1000, and told someone that you had done so, would the person know anything about the last n-1 of n digits (in the following example, the last 4 of 5 digits), or would it be random? For example, would the prime end in [...]0000x through [...]9999x with equal distribution? Or, could they know that, for example, ending in 9999x was far more likely than ending in 0000x? Thank you. 84.153.204.215 (talk) 17:24, 28 October 2010 (UTC)

sorry if it was unclear. My question can also be phrased like this. "Is it a good random number generator to find a prime of a thousand digits, and the first such prime you find, use its last n digits, discarding the last one first? For example, if you need a random number 0-10, you would use the second-to-last digit, and if you need a random number 0-100, you would use the third-to-last and second-to-last digits (so if the prime ends in ...563, then you get '56' as your random number). "84.153.204.215 (talk) 17:26, 28 October 2010 (UTC)

That would depend a lot on how you are generating your prime. If "the first such prime you find" is the same one every time, then it won't be random at all. If you have some really good way of generating the primes randomly, you can probably just generate random numbers directly anyway. Staecker (talk) 21:29, 28 October 2010 (UTC)

I'm guessing that the OP might be more interested in the distribution of digits in prime numbers than practical methods of generating random numbers? —Preceding unsigned comment added by 86.173.172.30 (talk) 21:40, 28 October 2010 (UTC)

How do you pick a 1000-digit prime? Doesn't it take a lot of work to figure out whether a specified 1000-digit number is prime? If you can already pick a 1000-digit number at random, then, given what you seem to want, why not stop there instead of worrying about whether it's prime or not? Michael Hardy (talk) 22:50, 28 October 2010 (UTC)

Asymptotically when n tends to infinite, Dirichlet's theorem on arithmetic progressions#Distribution says the digits have the same frequency. But for a fixed n, consider that the primes thin out on average per the prime number theorem. So 1000-digit primes starting with 1 are expected to be a little more common than those starting with 9. Starting with 10 should be slightly more common than starting with 19, and so on. And since starting with 20 should be more common than 29 and so on until 90 and 99, we also expect a second digit of 0 to be more common in total than a second digit of 9. If you ignore the first few digits then the difference becomes negligible for most purposes. But as others have pointed out, if you have a method to select a large random prime then you should be able to select a large random number without wasting time on prime generation. And if your prime isn't random then things depend on how the prime was selected. By the way, there are approximately 3.9×10⁹⁹⁶ 1000-digit primes. PrimeHunter (talk) 23:52, 28 October 2010 (UTC)