Are there any. I do seem to recall one equation or something that was proving difficult to solve, but I can’t think of the name at the moment.
Is there any calculation that our boffins and their computers can not come up with a solution for, currently?
I don’t exactly understand what you’re asking for, but the Ackermann function might be a candidate.
I vaguely remember something about whether Pi is a prime number being the subject of conjecture in the maths world. Although I could be wrong…
Prime numbers are a subset of the natural numbers, which are made up entirely of integers. Pi most assuredly is not an integer and so the concept of whether it is prime does not apply.
As you’ve phrased it, you’re absolutely wrong.
Pi cannot be prime, as it is not an integer. Pi is an irrational number–that is, it cannot be expressed as the ratio of two integers (this makes it distinct from, say. 2.5, which is the ratio of 5 to 2.) Put another way, pi is an infinite and non-repeating number; though it has a definite value, it can be written endlessly without any pattern ever emerging in the numbers after the decimal. It is also transcendental, that is, it is not a solution of a non-zero polynomial equation with rational coefficients.
But it sure ain’t prime.
Since prime numbers must be positive integers, and pi is definitely not an integer, there’s not much room for conjecture there.
Ahhhhhhh. Oops.
Just what it says on the tin, really. Are there any mathematical problems that have yet to be solved?
If anyone could provide me with a pocket history of mathematical milestones, with as few equations as possible, I’d be very appreciative. 
There are numerous major problems, thousands of minor problems, and potentially an infinite number of problems that can’t be proved with today’s math.
None of them are equations. Math doesn’t solve equations, it states theorems and then proves them.
Some of these are expressed extremely simply.
Goldbach’s conjecture: Every even integer n greater than two is the sum of two primes.
Riemann hypothesis: all non-trivial zeros of the Riemann zeta function have real part 1/2.
They still have defeated all mathematicians for hundreds of years.
If they are proven, they will immediately prove dozens or hundreds or thousands of hypotheses that depend on their validity for their own proofs.
But that will immediately set off a new round of a even greater number of new conjectures.
Math is infinite. We conquer it a tiny piece at a time. Mathematicians continually invent (or discover, depending on your point of view) whole new branches of math all the time. Math cannot, even in theory, be completely solved. (A proven theorem states that some results can never be proven within the branch of mathematicians that they occur.)
It’s a common misconception that mathematicians solve equations and that they know the answers to everything. Math doesn’t work that way at all. What you’re thinking of is the arithmetic you were taught in schools. Real math has about as relation to arithmetic as the internet has to a handheld calculator.
Depends on what you mean by “unsolvable mathematics problems.” There are some things that have been shown to be impossible. For example, since pi is irrational, the problem “Find a fraction (ratio of whole numbers) equal to pi” would have no solution.
Famous problems of this sort include squaring the circle, trisecting an angle, and finding an algebraic solution to the general quintic equation (i.e. finding a “quintic” formula for solutions to fifth degree polynomial equations akin to the quadratic formula for second-degree equations).
Then, there a number of famous (and not-so-famous) unsolved problems in mathematics, such as the “Millenium problems.” They’re only “unsolvable” until someone manages to come up with a solution. Although, thanks to Godel’s Incompleteness Theorem, we know that there are true statements that cannot be proved, though of course we don’t know which ones they are.
This depends a lot on what you mean by “solved.”
Linear equations a + b*x = 0 can be solved analytically for x: x = -a/b.
Quadratic equations ax^2 + bx + c = 0 can be solved by the quadratic formula. Similar formulae apply for the general cubic and quartic equations. But it has been proved that no such general expression is possible for 5th and higher powered polynomial equations. (Some can be factored of course, but there is no general formula that works in all cases.)
However all such equations can be solved to as much accuracy as desired for any specific case (specific real values for the coefficients.
If you’re looking for something that we know less about the solutions, I’d propose the Reinmann zeta function
ζ(s) = 2^s π(s-1) sin(πs/2) Γ(1-s) ζ(1-s)
also defined by the infinite series"
ζ(s) = 1^(-s) + 2^(-s) + 3^(-3) + …
The equation ζ(s) = 0 has solutions -2 -4, … These are the so called “trivial zeroes” of the function. If you let s be a complex variable, though, there are other (infinitely many) non-trivial zeros or solutions.
The famous Reinmann hypothesis is that all non-trivial (complex) zeros are of the form (1/2 + y*i). Whether or not this is true is one of the most famous unsolved problems in math. This conjecture is also related to many seemingly completely unrelated topics in math like the distribution of prime numbers.
But note that the OP didn’t initially use the words solved or unsolved (though his later clarification did, thereby confusing the issue a bit). The original question was whether unsolvable problems exists.
The answer is yes. A classic case is the Turing machine halting problem - it has been proved that a solution cannot be found.
No it wasn’t. He said, “Is there any calculation that our boffins and their computers can not come up with a solution for, currently?”
The modifier “currently” clearly, to me at least, indicates he means unsolved not unsolvable.
Thanks for the responses. Godel’s Incompleteness Theory was the one I was trying to remember.
Well, if you actually mean “unsolvable” rather than “unsolved”, then most definitely yes, as dictated by something called the Goedel’s Incompleteness Theorem. In very simple terms, it states that given a sufficiently complex system, there are always going to be true statements that cannot be proven true (and conversely false statements that cannot be proven false). Which also means there are “unsolvable” problems. In fact, there’s a neat proof for certain systems that the set of all possible problems is uncountable, but the set of all possible solutions is countable, so the probability of any random problem having a solution is 0.
Solvable but unsolved, on the other hand, is a more murky topic. There is very few (if any) things that we can prove to be solvable, without effectively solving them as part of that proof. Hence, most interesting unsolved problems are not yet known to be solvable.
Edit: This teaches me to hit “Reply” before lunch and then finish it after, without previewing.
Is that correct. I thought there were only true statements that were not provable. I thought the reasoning went roughly. If a statement is false there must exist some counter example. But the counterexample in and of itself is the proof that it’s false.
You’re limiting yourself to a very particular kind of statement. Clearly, if A is true and can’t be proved true, then the negation of A is false and can’t be proved false.
There is something called a Sigma_1 statement, which has the form “there exists an X with property P”, where one is able to enumerate through all possible Xs and check whether or not they actually have property P one at a time. If a Sigma_1 statement is true, then it is necessarily provably true, but it can be false without being provably false. Conversely, the negation of a Sigma_1 statement is called a Pi_1 statement; it has the form “every X has property P”, where, again, one is able to enumerate through all possible Xs and check for property P one at a time. Clearly, if a Pi_1 statement is false, then it is necessarily provably false, in the manner you were suggesting, but a Pi_1 statement may be true without being provably true.
Goedel’s Incompleteness Theorem guarantees that, fixing any particular formal system of a certain kind and also fixing an intended interpretation by which to judge the formal system’s sentences as true or not true, there is a true (on the intended interpretation) Pi_1 statement which the formal system doesn’t prove true; the negation of this will be a false Sigma_1 statement which the system doesn’t prove false.
Eh, depending on how you use your terms, this is either trivial (because everything solvable is to be considered already solved) or false (because some things are known to be solvable, without the solution having already been carried out).
It’s known that either white can force a win in chess, black can force a win in chess, or both sides can force non-losing in chess, and it’s also known that we could, in some sense, solve this problem by brute force. Yet all the same, that calculation will take a long, long, long time, and no one knows the answer…
I think it is an open question as to whether pi is a normal number Normal number - Wikipedia - most mathematicians think it is, but there’s no proof.
I’ll just bring in NP-Complete problems. They’re not “unsolvable” per se, but they can’t be solved except through brute force.