What are some "deep" math problems?

There are the Clay Institute Millennium Prize problems, with a cool $1M prize for solving. Of the 7 problems, 6 remain unsolved. (The Poincare conjecture was proved by Grigori Perelman, who refused the prize in 2010 and also refused the Fields Medal awarded in 2006.)

1 P versus NP
2 The Hodge conjecture
3 The Poincaré conjecture (proven)
4 The Riemann hypothesis
5 Yang–Mills existence and mass gap
6 Navier–Stokes existence and smoothness
7 The Birch and Swinnerton-Dyer conjecture

Although I’m a fan of mathematics, I have only a dim notion of what some of these are about, and of others I am clueless. #1 is related to computer science, while 5 & 6 are related to physics. The others are “pure math” with seemingly no real-world applications, though history shows that these sometimes later turn up at the core of new physical theory.

The Collatz Conjecture. The famous peripatetic mathematician Paul Erdős said of the Collatz conjecture: “Mathematics is not yet ready for such problems.”

@gasman The Reimann hypothesis has to do with the pattern between prime numbers, which has very significant cryptographic applications. Since in the technological era, data is encoded with numbers, and every number is made up of prime numbers. Knowing the “key” to prime numbers would allow much more efficient cracking of digital codes.

https://secure.wikimedia.org/wikipedia/en/wiki/Knights_and_Knaves

See if you can figure them out without looking at the solutions.

@Neophyte: Indeed – and factoring large numbers (hundreds of digits) is practically impossible with present-day methods – at least until quantum computing comes to full fruition. The irony is that number theory, the “queen of mathematics,” was long regarded as the last bastion of pure math with no practical applications – until public-key cryptography came along in the late 1970s.

A good read on the Riemann hypothesis is Prime Obsession by John Derbyshire.

@Paradox1: Raymond Smullyan’s books have long been a source of entertaining and clever logic puzzles.

I posted this before – Peg solitaire and group theory It is a clever application of group theory. Assuming you are familiar with group theory, skip down to the section labeled Klein 4 trickery. It is non-intuitive in the sense that you specified though not very difficult.

@Paradox1 Question 1 and 2 were very easy. With question 3, I have heard of a different form of that question, so I already know the answer, and it doesn’t really count as me solving the problem.

Are there sets of cardinality strictly between that of the natural numbers and the real numbers? This is the Continuum Problem, posed by Georg Cantor in 1877. His hypothesis was that no such sets existed, but he never came up with a proof. Legend has it that this contributed to his struggle with mental illness.

David Hilbert included it in his famous list of 23 important unsolved problems in mathematics published in 1900, giving it the #1 slot.

In 1940, Kurt Godel proved that Cantor’s hypothesis could not be disproved from Zermelo-Fraenkel (ZF) set theory, which is generally regarded as the “gold standard” of axiomatic set theories. In 1963, Paul Cohen proved that Cantor’s hypothesis could not be proved from ZF set theory either. This would seem to indicate that “everything” we know about sets is still insufficient to determine the answer to this problem.

As set theory is considered to be the very foundation of mathematics itself, this gives it a serious claim to being the “deepest problem”, which is why Hilbert put it at the head of his list.

@PhiNotPi Very well then!

Now without looking it up tell me the complete value of phi along with all of the practical applications it has in our society… just kidding

@Paradox1 The link on that page to “the hardest logic problem in the world” was very interesting, and was very difficult. But without looking up, I am not very sure of the value of phi. Is it (1 + sqrt(5))/2? Practical applications include architecture and painting, and other visual arts, where it is commonly seen as a golden rectangle.

<hopes I’m right>

Take the coordinates of the Sun, Earth and Jupiter right this very second and apply Newton’s law of universal gravitation which states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

@PhiNotPi The first three digits are .618, as Phi is known as “The Golden Ratio.” I had to study it in my finance courses as it relates to investing in the stock market. The theory is that when prices react they react to Phi levels. The book “Nature’s Law” espouses that all market price patterns are due to the fact that humans think and behave in this ratio. I have to say this sounds crazy, though it often is uncanny in that it seems to work. No one has really proven it one way or another which is why I say seems.

Phi can be simply derived by taking Fibonnaci sequence and dividing N by N+1, starting from 3. 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. The further along the sequence you get the closer the actual decimal approaches phi. For practical purposes, .618 will do, as will its inverse, .382.

@Paradox1 The value of phi is actually 1.618…

Touche good sir, touche. I am so used to calculating it as a percent for practical applications I had forgotten.