The twenty-five mathematical problems and questions listed below were first posted on the internet in 1995. Since that time, Problems 5, 7, 8, and 22 have been solved completely, and part of Question 12 has been answered. The other problems remain unsolved. The links in this list point to articles on the MathPages web site containing more background on each problem, and partial or related results.

(1) If each "1" in the binary representation of the integer x signifies a point in the corresponding position on a linear lattice, and if x' denotes the binary digit reversal of x, prove or disprove that the equality xx' = yy' implies that x and y have the same multi-set of point-to point distances. Ref: Generating Functions for Point Set Distances Isospectral Point Sets in Higher Dimensions

(2) Find an elementary proof that x^2 + y^2 and x^2 + 103y^2 cannot both be squares for non-zero integers x,y. Ref: Concordant Forms

(3) Prove (or disprove) that the only solutions of ab = c (mod a+b) ac = b (mod a+c) bc = a (mod b+c) in positive coprime integers are (1,1,1) and (5,7,11). Ref: A Knot of Congruences Limit Cycles of xy (mod x+y) More Results on the Form xy (mod x+y)

(4) Given any cyclical arrangement, A, of |G| elements (not necessarily distinct) of a group G, let f(A) denote the cycle consisting of the compositions of the cycles of A. If A consists of all |G| distinct elements of G, is it true that iterated application of f eventually leads to the identity cycle (i.e., the cycle consisting solely of the identity element of G)? Ref: Permutation Loops Cumulative Permutation Sequences String Algebra

(5) In how many distinct ways can the integers 0 through 15 be arranged in a 4x4 array such that the bitwise OR over each row, column, and diagonal is 15, and the bitwise AND over each row, column, and diagonal is 0? Ref: 414298141056 Quarto Draws Suffice! Ref: Four Cubed SOLVED: 12 Dec 96. See the article 414298141056 Quarto Draws Suffice! for the answer to this combinatorial question, and a description of how it was solved. Special thanks to Steve Zook for being the first to settle one of the original 23 Most Wanted problems. Thanks also to Andrei Ivanov for independently confirming the result, and to Keith Lynch for useful discussions on this topic.

(6) Is it true that ln(x)^2 | pi(x) - g(x) | is less than ---------- 4 where pi(x) is the number of primes less than x, and g(x) is the number of composite integers n less than 2x such that n plus the sum of its prime factors (counting multiplicities) exceeds 2x? Ref: Sum of Prime Factors(M)

(7) Let M(n,k) denote the number of distinct monotone Boolean functions of n variables with k mincuts. Obviously M(n,0)=1 and M(n,1)=2^n. In addition, we have M(n,2) = (2^n)(2^n - 1)/2 - 3^n + 2^n M(n,3) = (2^n)(2^n - 1)(2^n - 2)/6 - 6^n + 5^n + 4^n - 3^n Is there an explicit formula for M(n,k) with k=4? Ref: Dedekind's Problem Progress on Dedekind's Problem SOLVED: 17 Sep 99. I've received an email from Vladeta Jovovic informing me that he, G. Kilibarda, and Z. Maksimovic of the University of Belgrade are preparing a paper in which they give explicit expressions for M(n,k) with k=4 to 10. See Progress on Dedekind's Problem.

(8) The number 2^(k-1) + k is a prime for k = 1, 3, 7, 237, and 1885. Does anyone know of any other primes of this form? Ref: Sum of Prime Factors(M) ANSWERED: 10 Nov 99. Nuutti Kuosa has found that 2^(k-1) + k is a prime for k = 51381. For more details, see the note Sum of Prime Factors(M).

(9) For any positive integer N let m_k, k=1,2,...t denote all the solutions of m+SOPF(m)=N, where SOPF is the sum of prime factors (including multiplicities). Then define t tau(N) = N - SUM SOPF(m_i) i=1 Is tau(N) equal to zero for any integer N? Ref: Sum of Prime Factors(M)

(10) Let c(N) denote the number of distinct configurations of N particles in static equilibrium on the surface of a sphere of radius R, assuming the particles repell each other with a force that varies as 1/(1+r^2). How does c(N) vary (if at all) with R? Ref: Background for Problem 10

(11) For equilibrium configurations (EC) of N particles on a sphere under the influence of an inverse-square repulsive force law, there is an EC for N=17 that contains an EC for N=7 as a subset. Also, there is an EC for N=22 that contains an EC for N=6 as a subset. Also, there is an EC for N=23 that contains an EC for N=5 as a subset. Are there any other (non-trivial) examples of this kind? Ref: Points On A Sphere

(12) For any prime p let z(p) denote the number of distinct solutions of x^2 x^3 x^(p-1) 1 + x + --- + --- + ... + ------- = 0 (mod p) 2! 3! (p-1)! Is it true that the fraction of primes with z(p) = n asymptotically approaches a Poisson distribution, i.e., the fraction of primes p less than x with z(p)=n approaches 1/e*n! as x goes to infinity? Also, what is the smallest prime p such that z(p)=6? Ref: A Special Property of 151 PARTIALLY ANSWERED: 17 Jun 00. The second part of Question #12 has been answered. It asked for the smallest prime p such that the generalized exponential function has 6 roots modulo p. Don Reble has found that p = 11117 is the smallest such prime. Also, he evaluated all the primes less than 32768 and found the overall distribution in good agreement with the conjectured Poisson density. See the updated note on A Special Property of 151 for more details.

(13) For primes p of the form 3k-1 the congruence x^p + y^p = (x+y)^p (mod p^2) has a solution with x,y,(x+y) all non-zero (mod p) iff p is one of the primes 59, 83, 179, 227, 419, 443, 701, 857, 887, 911, 929, 971, 977,...etc. Is there a simpler way of characterizing these primes? What is their density? Ref: On the Density of Some Exceptional Primes

(14) The number 588107520 is expressible in the form (X^2 - 1)(Y^2 - 1) (where X,Y are integers) in five distinct ways. Is there a 6-way expressible number? Ref: Numbers Expressible As (a^2 - 1)(b^2 - 1)

(15) For what integers k and n do there exist integers x,y,z such that |x|,|y|,|z| > k^(1/n) and x^n + y^n + z^n = k? (With k=0 this is just the Fermat problem.) Ref: Marginalia: A Conjecture On The Fermat Function Ref: Sums of Three Cubes

(16) With k=1,n=3 in problem (15) above, an infinite family of solutions is given by (1 +- 9m^3)^3 + (9m^4)^3 + (-9m^4 -+ 3m)^3 = 1 but this doesn't cover all the solutions with k=1,n=3. Are any (all?) of the other solutions given by an algebraic identity? Ref: Sums of Three Cubes

(17) For any positive integer n let SOPF(n) denote the sum of the prime factors of n, including multiplicities. Is it true that every iteration of the form x -> SOPF(ax+b) for constants a,b is ultimately periodic? Is the number of limit cycles finite for any given a,b? Sum of Prime Factors(M)

(18) With SOPF(n) as defined in (17) above, are there infinitely many solutions of SOPF(x^2 + ax + b) = x for any integers a,b? Are there solutions of SOPF(x^2 +1217x + 370313) = x such that the quadratic is NOT the product of three primes? Is there a relation between the class number of b^2 - 4a and the number of solutions of SOPF(x^2+ax+b)=x? Is there a relation between the period of cycles of SOPF(x^2+ax+b) and the class number of b^2-4a? Sum of Prime Factors(M)

(19) Given any circular arrangement of the n integers j through j+n-1, let S denote the sum of the qth powers of every sum of k contiguous numbers. Then let v(q,k,j,n) denote the number of distinct possible values of S for all possible arrangements. Can v(q,k,j,n) be expressed in closed form as a function of the indices? Which integer sequences are contained within this family? Which continuous functions (e.g., sin(x), exp(x)) are approximated by members of this family? Ref: The Dartboard Sequence

(20) Suppose I announce a sequence of binary digits beginning with 1, such as "1 1 0 1 ..." representing the numbers 1, 3, 6, 13, and so on. Your objective is to stop me at some point and, by supplying one more binary digit, produce a number divisible by at least one of a given set T of "target" primes. For any given x>0, does there exist a set T of primes p>x such that you are guaranteed a winning opportunity? What is the minimum number of elements of such a set? Ref: Binary Games

(21) Let a(n) and b(n) denote integer sequences each satisfying the recurrence s[k] = 4s[k-1] + 9s[k-2] with the initial values {1,0,...} and {0,1,...} respectively. Find a composite integer N congruent to 2,5,6,7,8, or 11 (mod 13) such that a(N)=4 and b(N)=-1 (mod N). Ref: Pseudoprimes For x^2 - 4x - 9 Lucas and Perrin Pseudoprimes Symmetric Pseudoprimes

(22) Given two integer sequences X = {x1,x2,...} and Y = {y1,y2,...}, let X = P(Y) signify that xn is the number of partitions of n into elements of Y. The two sequences X = {1,2,3,4,6,9,11,15,19,25,31,41,49,61,75,91,110,...} Y = {1,2,3,5,6,10,12,17,22,29,36,48,58,73,91,111,...} are duals of each other in the sense that X=P(Y) and Y=P(X). Do there exist three sequences X,Y,Z such that X=P(Y), Y=P(Z), and Z=P(X)? Ref: Enumerating All Cycles of Partition Sequences SOLVED: 19 Jan 97. See Enumerating All Cycles of Partition Sequences for Dan Ford's analysis of this problem, and some additional generalizations and questions.

(23) For any positive integer n let tau(n) and sigma (n) denote the number and sum of the divisors of n, respectively. A number N is called "sublime" if tau(N) and sigma(N) are both perfect (in the ancient Greek definition). The only two known sublime numbers are 12 and 6086555670238378989670371734243169622657830773351885970528324860512791691264 Can anyone find another sublime number, or prove that no others exist? Can there exist an odd sublime number? Ref: Sublime Numbers

(24) Prove or disprove that no sum of two or more consecutive positive nth powers equals an nth power for any n>3. Ref: Sums of Consecutive Nth Powers Equal to an Nth Power

(25) Prove or disprove that there do not exist constants A,B and integers xi,yi (i=1,2,3,4) such that yi^2 = A xi^2 + B i = 1,2,3,4 and such that the four products (xi)(yi) are in arithmetic progression. Ref: No Progression of Four Rectangles on a Conic?

Return to MathPages Main Menu