## MathPages Wanted List

```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

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
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

(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?
```