Clifford A. Pickover

I look forward to hearing from you. Please e-mail me your answers and I will post information here. (E-mail address is on main page.)

1. What are the five most famous or important unsolved mathematical
problems today?

2. What is the most difficult-to-understand areas of mathematics
today?

3. Who are the five most influential mathematicians that ever lived?

4. Who are the five most influential mathematicians alive today?

5. Who was the strangest mathematician that ever lived?

6. What are the five most interesting numbers? (And 5 most interesting
integers?)

7. What one question would you like seen added to this list?

**
Note: these questions and many more are now answered in my book
Wonders of Numbers!
**

To question 3, who are the 5 most influential mathematicians of all time, here are my answers: A) Gauss, B) Euclid, C) Newton, D) The Bernoulli's, E) Descartes, As for question 2, the most-difficult-to-understand area of mathematics today, I would venture to say that F) Advanced Number Theory is the hardest; ANT includes the mathematics used in the proof of Fermat's Last Theorem (by Wiles). As for question 4 (most influential living mathematicians), here's a partial list of my answers: G) Andrew Wiles H) Stephen Wolfram I) Chudnovsky Brothers and their computer, m-zero As for question 1, the most important unsolved problems, here's a partial list also: J) Proof of the Riemann Hypothesis K) A true pi(n) function (where pi is the symbol pi; that function is the prime number function in number theory. Currently, to know the number of primes up to n, we have to count them "by hand", using a computer. However, some people think that there might be a function that can truly determine the number of primes up to n.) As for Question 6, the 5 most interesting numbers, L) pi M) e^(i*pi) [ = -1 ] N) The Golden Ratio, phi, and phi' which is

1 + sqrt(5) 1 - sqrt(5) ----------- and ----------- 2 2O) Skewes' number. P) The current largest Mersenne Prime, M(3021377). As for the others, I'm still thinking. (5) is particularly hard, since many mathematicians can be considered "strange". As for question 7, I'd like to see the question "What is the most important equation", or even "What are the five most important equations" for which I feel that e^(i*pi) + 1 = 0 is the most important because it connects e, pi, i, 1 and 0 all together. Thus it connects calculus (e), geometry (pi), number theory and complex numbers (i), and arithmetic (1 and 0) all in one shot. That's impressive! 1 + (1/32) + (1/52) + (1/7) ... = pi2/8 is also important; but the equation:

lim e^n * n! n->inf. ------------- = sqrt(2 * pi) n^n * sqrt(n)is of amazing importance. Also, phi^n = phi^(n-1) * phi^(n-2) is quite important. Probably the 2nd most important equation (after e^i*pi + 1 = 0), is this limit:

lim sqrt(n + sqrt(n +...)).. = 1 n->0That is, the continued square root of 0 is equal to 1, even though the square root of 0 is 0.

The 5 most interesting integers: 0 , 1 (the only two numbers needed to start the sequence of integers). e^i*pi, which is an integer = -1, thus making -1 the only integer that can be expressed in terms of those three non-integer numbers (e, i, pi). 2 (the only even prime) The largest Mersenne Prime, because it contains over 900,000 digits and yet was tackled with common PC's, not vector computers.

From Olivier Gerard:

1. A) The Riemann Hypothesis, B) The Goldbach Conjecture, C) The
normality of Pi digits in an integer base, D) Kepler Conjecture, E)
Poincar=E9 Conjecture.

2. A) Quantum Groups, B) Motivic Cohomology, C) Infinite Banach
Spaces, D) local and micro local analysis of large finite groups, E)
Large and inaccessible cardinals

3. Euler, Gauss, Hilbert, Dedekind, Galois

4. Vladimir Arnol'd, John Conway, William Thurston, Alain Connes,
Ed Witten 5. Ramanujan 6. 0, i, 2, Euler's Gamma, Chaitin's Omega,
Aleph 1

I think all that requires some comments: I have deliberately made
very "classic" choices for most questions especially 1,3,5. The order
is most of the time meaningless. The answers someone could make to
this questionnaire are more revealing of his mathematical culture,
background and activity than anything else.

1. My list is very "classic" in the sense that most of these
problems were already posed before 1900. There is an ongoing study to
write out the most important math problems for the 21st century, in
the frame of the forthcoming World Math Year 2000. I think this is
under the auspices of the IMU.

2. The question is inevitably biased. A theory can be difficult
because it is awfully written and notated. It can also be temporarily
difficult because some pieces are lacking (such as in a puzzle) or a
good synthesizer has not yet put some order in several layers of
recent or less recent ciphering themselves mutually. I have very
rarely seen genuine or eternal difficulty in mathematics, but often
conjectural or cultural one. My answer A), B) and D) are difficult
because you must go through a lengthy initiation, review and training
process before hoping to say anything useful or slightly new and at
present they seem to mainly feed themselves. C) and E) are difficult
because off the well-traveled roads of progressive intuition building
of "mainstream" mathematics but if you like what others could call
"pathology" or "nonsensical" exotism, they have the advantage of being
quite stimulating.

5 items is perhaps not a comfortable bound for most topics. One
of the trouble for instance with question 4 is related with the Fields
Medal. It cannot be presented to more than 4 people at a time because
its renown would be diluted, but in the same time, at each
International Congress, a good dozen are really worth it. Influential
people are less common, but for 4, very influential people are still
alive but quite old and not so active as they were. Example:

4old: Henri Cartan, Ren=E9 Thom, B.L. Van der Waerden, Saunders
MacLane, (Eilenberg is dead this year), (Gorenstein, the leader of the
attack of the classification of simple groups is dead a few yeas ago)
Itoh, Andr=E9 Weil, Walter Feit, Gustave Choquet...

4bis: Eugen Dynkin, Jean-Pierre Serre, Donald Knuth, Stephen
Smale, Ingrid Daubechies, Doron Zeilberger, Vladimir Drin'feld, Pierre
Cartier, Herbert Wilf, George Andrews, B. Mandelbrot, Gerd Faltings,
Martin Kruskal, Armand Borel, Jean-Yves Girard, Lawvere, Harvey
Friedmann, Andrew Odlyszko, Philippe =46lajeolet, ...

Other World Class Mathematicians:
3bis: Riemann, John von Neumann, Leibniz, Henri Poincar=E9, Allan
Turing, Niels Abel, Sophus Lie, Felix Klein, Lagrange, Fermat, Euler,
Grothendieck, Carl Siegel, Jacobi, Camille Jordan, Cantor, Hadamard,
Hardy, Littlewood, Newton, Girard Desargues, Pascal, Weierstrass,
Gentzen, Tschebycheff, Markov, Kolmogorov, Fredholm, Banach, Cayley,
Sylvester, Mobius, Cauchy, Laplace, ...
(note Grothendieck is still alive but no more mathematically active)

5. Ramanujan is the strangest but at the same time has been much
studied. In many ways, the isolated mathematical genius is a source
of wonder but when mathematicians live long like Newton, Gauss or
Euler, the later achievements remove the drama of their beginnings.
The birth of the mathematical abilities of Abel and Galois is not
easily explained. The only common factor seems to be a tremendous
amount of time and energy devoted to their passion, and a relative
intellectual independance.

6. I have put 0 and 2 because I feel there is still much to
understand in the process of counting and its ramifications, the
notion of an origin, etc.= =2E i is there for these reasons but it is
also the symbol of the algebraic closure of reals, a fantastic
"passeur" and tool for many fields (as in number theory), and an
historical problem of representation. Euler's Gamma is another kind
of link : between the exponential/logs on a si= de and number theory
on the other. This is one of the best representatives of these
recurrent constants which grows a mystical aura. Chaitin's omega
shows among other things the contrast between the wide expression
power of human (natural or artificial) languages (that we use in
definitions), and the very limited machinery of our mathematics (that
we use for demonstrations). Aleph 1 is the symbol of the central
independence results of set theory, (AC, CH), one of the biggest
mathematical events of the 20th century.

Other questions could be:
7n) What are the five mathematical theories you find over-rated or
too publicized to be easy to work seriously in or sort the hype ?
8n) What are the five mathematical theories you find under-rated,
unknown, underused, ... ?
9n) What are the five mathematicians who do best when
communicating their fields and results to non-specialists ?
10n) What are the five (mathematicians or not) who do best when
communicating their results (or other's) to non mathematicians ?
11n) Can you quote 5 living mathematicians whose writing or oral
skills are particularly poor ?
12n) What are the 5 five most boring question you get asked on mathematics ?

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