A collection of easily stated number theory conjectures which are still open. Each conjecture is stated along with a collection of accessible references.

- The Riemann Hypothesis
- Perfect & Mersenne Numbers
- The Twin Primes Conjecture & Prime Gaps
- Fermat Numbers
- Goldbach's Conjecture
- Catalan's Conjecture
- Totient Function Conjectures
- The Collatz Problem

__Def:__ Riemann's Zeta function, Z(s), is defined as the
analytic extension of
_{n=1}^{infty}1/(`n`^{s})

__Thm:__ `s`)=prod_{i=1}^{infty}
(1-(1/`p`_{i}^{s}))`p`_{i} is the `i`th prime.

__Thm:__ The only zeros of Z(`s`) are at `s`=-2, -4, -6, ...
and in the strip 0<Re(`s`)<1.

__Conj:__ The only zeros of Z(`s`) are at `s`=-2, -4, -6, ...
and on the line Re(`s`)=1/2.

__Thm:__ The Riemann Conjecture is equivalent to the conjecture that
for some constant `c`,
`x`)-li(`x`)| < `c`sqrt(`x`)ln(`x`)`x`) is the prime counting function.

*Unsolved Problems*, Klee & Wagon, 1991, Sect 17*Elementary Number Theory*, Jones & Jones, 1998, Chap 9-
`http://www.utm.edu/research/primes/notes/rh.html`

(notes in Chris Caldwell's Prime Pages) `http://match.stanford.edu/rh/`

(Dan Bump's notes)

__Def:__ `n` is *perfect* if it is equal to the sum of
its divisors (except itself). Examples are 6=1+2+3, 28, 496, 8128, ...

__Def:__ The `n`th Mersenne Number, M_{n},
is defined by _{n}=2^{n}-1

__Thm:__

- M
_{n}is prime implies that2 is perfect. (Euclid)^{n-1}(2^{n}-1) - All even perfect numbers are derived from Mersenne primes in this way. (Euler)

__Conj:__ There are infinitely many Mersenne primes (equiv perfect
numbers) and infinitely many Mersenne non-primes.

__Conj:__ There are no odd perfect numbers.

The search for Mersenne primes is a hobby that anyone can participate in. Assistance, code and results are available from GIMPS (the Great Internet Mersenne Primes Search).

*Unsolved Problems*, Klee & Wagon, 1991, Sect 16*Number Theory*, Shanks, 1985, Chap 1*Elementary Number Theory*, Jones & Jones, 1998, Sects 2.3 & 8.2-
`http://www.utm.edu/research/primes/mersenne.shtml`

(notes in Chris Caldwell's Prime Pages)

__Def:__ *Twin primes* are a pair of primes of the
form {`n`, `n`+2}. Examples are
{`n`, `n`+2} =
{3, 5}, {5, 7}, {11, 13}, {17, 19}, {29, 31}, ...

__Conj:__ There are an infinite number of twin primes.

__Def:__ A prime gap is the difference
`p`_{i+1}-`p`_{i}`p`'s are consecutive primes.

Various conjectures about the distribution of prime gaps are current. Tables of the earliest occurrence of each prime gap up to at least 804 have also been tabulated (and can be added to). There is also a distributed search for larger twin primes which has a page at http://www.serve.com/cnash/twinsearch.html.

*Unsolved Problems*, Klee & Wagon, 1991, Prob 16.1*Number Theory*, Shanks, 1985, pp30*Unsolved Problems in Number Theory*, Guy, 1994, Sect A.8*The Book of Prime Number Records*, Ribenboim, 1989, Sect 4.III- Chris Caldwell's Prime Pages:

__Def:__ The `n`th Fermat number, F_{n},
is defined is _{n}=2^{2^n+1}

__Conj:__ Only a finite number of Fermat numbers are prime.
A stronger conjecture is that the only prime Fermat numbers are
F_{0}=3, F_{1}=5, F_{2}=17, F_{3}=257,
and F_{4}=65537.

*Number Theory*, Shanks, 1985, Chap 2*Unsolved Problems in Number Theory*, Guy, 1994, Sect A.3*The Book of Prime Number Records*, Ribenboim, 1989, Sect 2.VI

__Conj:__ `n` even and greater than or equal to 4 implies that there
are primes `a` and `b` such that
`n`=`a`+`b`

*Number Theory*, Shanks, 1985, pp30*Unsolved Problems in Number Theory*, Guy, 1994, Sect C.1*The Book of Prime Number Records*, Ribenboim, 1989, Sect 4.VI-
Verifying Goldbach's Conjecture up to 4 x 10
^{14}

__Conj:__ The only consecutive prime powers are 8=2^{3}
and 9=3^{2}.

Langevin and Tijdeman showed that any counterexamples must be smaller than exp(exp(exp(exp(730)))), so in theory it can be proved with a computation. This bound is so large that some major cutdown in possible counterexamples is still needed.

*Unsolved Problems in Number Theory*, Guy, 1994, Sect D.9*Catalan's Conjecture*, Ribenboim, 1994

__Def:__ Euler's totient function, phi(`n`) is the
number of integers less than `n` which are coprime to `n`.

__Conj:__ For no composite `n` does phi(`n`)
divide (`n`-1). (Such a number would be a Carmichael
number.) (Lehmer)

__Conj:__ (Carmichael's Conjecture) For all `m` there
is an `n` such that phi(`m`)=phi(`n`).

__Def:__ The *multiplicity* of `n` is the number
of integers `k` such that phi(`k`)=`n`.

__Conj:__ All integers greater than 1 have occur as multiplicities.
(Sierpinski)

*Unsolved Problems in Number Theory*, Guy, 1994, Sect B.37

Consider the function `f`(`x`)=

`x`/2 if`x`even- 3
`x`+1 if`x`odd}

__Conj:__ For any integer `n` there is an integer
`d` such that
`f ^{d}`(

This conjecture is variously referred to as the Collatz problem
(for the original worker in the field), the Syracuse problem,
or the `x`+1

*Unsolved Problems*, Klee & Wagon, 1991, Prob 19*Unsolved Problems in Number Theory*, Guy, 1994, Sect E.16- The 3x+1 Problem, 1985 paper by Jeff Lagarias, originally in MAA Monthly
- Notes by Eric Roosendaal

Robert Campbell Nov 29, 1998