Potpourri on Difficult/Unsolved Mathematics Quiz

Answer: Basel problem

The Basel problem asks for the value of the infinite sum of the reciprocals of every perfect square number greater than 0. This is exactly equal to (pi^2)/6, or about 1.644934, as proved by Euler.

This problem was first set forth in the early 1600s, and was solved in 1735 by means of extrapolating properties of finite polynomials to the Taylor series expansion of the sine function (with a few other manipulations), and although his logic was somewhat doubted due to logical questions that Euler couldn't sufficiently answer, his obtained answer through the method matched perfectly with calculations.

"Number Theory: An Approach Through History" by André Weil

Answer: Apéry's constant (~1.202)

Even if the name and the trivia was unknown, this can be solved using elimination.

With only a ballpark idea about the probability of triple coprime numbers (~83% realistically), one can eliminate Gelfond's constant, as the probability is drastically above 1/23.141, or ~4%.

The square root of 3 and similar have been proven irrational far before 1978 (the Pythagoreans, BC mathematicians, knew this).

The golden ratio is (1+√5)/2, which is 1/2+√5/2, which is easily identifiable as an elementary irrational number by the latter part, and an educated guess would say that it wouldn't take until 1978 to prove it as irrational. It would be correct; it was identified by Euclid in BC times as irrational.

"Prime Obsession" by John Derbyshire

Answer: Birch and Swinnerton-Dyer conjecture

The only defining difference between this and the Riemann hypothesis as described by the question is the reference to elliptic curves, which is a concept missing from the skeletal statements of all the other Millennium Prize Problems listed, including the Riemann hypothesis.

The Birch and Swinnerton-Dyer conjecture states that, denoting E as an elliptic curve function, the slope of an equation E(x) over a given number field K will tend toward the mathematical rank of E(K) as x approaches infinity (although only knowing the "elliptic curve" bit is enough for the question).

"Elementary Number Theory" by William Stein

Answer: Euler-Mascheroni constant

Pi is easily recognizable as irrational and transcendental.

Gelfond's constant (e^pi) is another constant, perhaps not as well known as pi, that has been proven to be irrational and transcendental.

The Riemann zeta function of 3 is actually Apéry's constant, which is proven irrational (although the quest is still on to prove the irrationality of many other integer Riemann zeta values).

The irrationality of the Euler-Mascheroni constant, the number denoting the sum of the reciprocals of the first x integers minus the integral of 1/ln(x) from 0 to x as x approaches infinity, is a difficult problem to solve, despite mathematicians' best efforts. Although, there is still the comforting fact that, if it were rational, the denominator is provably above 10^242080.

"Euler-Mascheroni Constant" from Wolfram MathWorld.

Answer: Infinitely many times

This was proved by Littlewood, much to everyone's surprise. There is no complete numerical evidence to say it crosses even once, and this is the reason that Gauss conjectured that the integral of 1/ln(x) is always greater, although Littlewood's logic is sound; indeed, this is where Skewes' numbers, the bounding numbers on this issue, arise, most famous for their stupidly large values, the likes of which have never been seen with meaning in mathematics before (such as 10^10^10^963).

"Prime Obsession" by John Derbyshire

Answer: Yes

150 is the area of a 15-20-25 triangle. A quick way to do this would be to notice that by multiplying 6, a two-dimensional area, by the perfect square 25 (and thus multiplying the one-dimensional side lengths by 5, and since rational*rational=rational, the resulting side lengths are rational), you obtain 150 as the area of a rational-legged right triangle.

"Elementary Number Theory" by William Stein

Answer: Harmonic series

The lower and upper bounds increase toward infinity since log2(x) diverges (slowly), so clearly it is not a geometric series with a ratio greater than 1.

The natural log of Pascal's Triangle's middle terms grows much too quickly, and starts too low. The first term (ln(1)=0) is below 1+log2(1)/2=1, and its fifth term (ln(70) is greater than 5) is greater than the upper bound 1+log2(5) which is greater than 4. Thus is doesn't fit within its prescribed area.

The primes-only harmonic series grows much too slowly, and with the first few terms it becomes clear that it is not above the lower bound (although note that it does diverge).

The divergent ever-growing harmonic series' (1+1/2+1/3+1/4+1/5...) alternating expression (1-1/2+1/3-1/4+1/5...) ends up perfectly converging to the natural logarithm of 2 (~0.693147) after an infinite number of terms. The harmonic series itself was first proved to diverge by a 14th century source, by finding the mentioned lower bound of 1+log2(x)/2, which clearly diverges. A similar method can be applied to obtain the upper bound of 1+log2(x).

"Elementary Number Theory and its Applications" by Kenneth Rosen

Answer: Poincaré's dodecahedral space

A Turing machine has no connections to topology in this question, Klein didn't form a signature torus to disprove Poincaré's results (although the two did indulge in a possibly less-than-healthy rivalry), and the Tipler cylinder is a hypothetical method of time travel through manipulation of spacetime curvature.

Poincaré's dodecahedral space, also called Poincaré's sphere, is also a conjectured shape of the universe, a hypothesis that gained attention after the WMAP spacecraft extensively studied cosmic radiation in the 2000 decade.

"The Poincaré Conjecture: In Search of the Shape of the Universe" by Donal O'Shea
"Is the universe a dodecahedron?" at Physicsworld

Answer: Collatz conjecture

The Collatz conjecture, proposed in 1937, has many other names (none of which were listed in the question), such as the Ulam Conjecture, Thwaites conjecture, Kakutani's problem, and more.

Brute-force computation has revealed that at least the first 5,000,000,000,000,000,000 (5*10^18) terms all spiral down to 1. Various other mathematical bounds have also been proved that limit the number of Collatz conjecture disproving numbers, such as, the number of terminating numbers between 1 and x is at least x^0.84, although the unproven conjecture holds that it is strictly x.

"Goldbach Conjecture" on Wolfram MathWorld

Answer: Odd

This is a question related to the Pólya conjecture, proposed in 1919 and counter-proven almost 40 years later in 1958. The first number that, for all numbers at or below it, have more of an even amount of prime factors than odd, is 906,150,257, figured in 1980. The Pólya conjecture hypothesized that such a number did not exist.

"Verschiedene Bemerkungen zur Zahlentheorie" (Several remarks on number theory) by George Pólya.

Answer: Riemann Hypothesis

Mertens conjecture and the fundamental theorem of arithmetic have already been proven false and true, respectively. Only the Riemann hypothesis, though, intrinsically links the truth of that certain pattern to itself. If the Riemann hypothesis is true, then so is the pattern, and vice versa. If the Riemann hypothesis is false, then so is the pattern, and vice versa.

"Prime Obsession" by John Derbyshire

Answer: e, pi, and i

e^(i*pi)+1=0 is gazed upon as it unifies e, a constant used extensively in finance and number theory, with pi, a constant used liberally in geometry, with i, the square root of negative one, used in various pure mathematics and electronics. Nowadays, the three intermingle in other ways to the degree that mathematicians aren't too surprised to see them pop up in unrelated equations, but back in the 1700s it was a sure surprise for them.

"Prime Obsession" by John Derbyshire

Answer: Maxwell's equations

Maxwell's equations, formally produced in the late 1800s, paved the way for modern advancements in electromagnetism as well as optics, and sped along theories on electricity. Two major variants of it exist, one for microscopic calculation, and one for macroscopic calculation, the latter of which is generally regarded as easier and less chaotic due to less interference from atomic influences.

"On Physical Lines of Force" by James Clerk Maxwell

Answer: Bernoulli numbers

Bernoulli numbers can be used, along with some of Pascal's triangle's properties, to form Bernoulli's theorem, which outputs the sum of powers up to a given number for a given power. It is also one of the first sequences for which an algorithm was able to be "programmed" to calculate it, specifically on Babbage's machine.

"Sketch of the Analytical Engine" by L. F. Menabrea.

Answer: Polignac's conjecture

Polignac's conjecture encompasses another well known problem in number theory; the twin primes problem. All of the listed numbers had gap size 2, and were thus twin primes. These are of particular interest in themselves to mathematicians. If Polignac's conjecture was proven true, then the twin prime problem would also be solved, which states there are an infinite amount of twin prime numbers.