Conjectures by Paul Erdős

The mathematician Paul Erdős made many conjectures in various areas of mathematics in his work .

Assumptions about number theory

Erdős-Moser conjecture : it says that the equation

{\ displaystyle 1 ^ {n} + 2 ^ {n} + \ ldots + (m-1) ^ {n} = m ^ {n}}

just the solutions and has.

{\ displaystyle (n, m) = (0.2)}

{\ displaystyle (1,3)}

Erdős-Straus conjecture : it says that the equation

{\ displaystyle {\ frac {4} {n}} = {\ frac {1} {a}} + {\ frac {1} {b}} + {\ frac {1} {c}}}

has a natural number solution for every natural number .

{\ displaystyle n> 1}

{\ displaystyle a, b, c}

${\ displaystyle \ {n \ in \ mathbb {N} | \ forall \; k \ in \ mathbb {N}: \; \; 2 ^ {k} <n \ Rightarrow n-2 ^ {k} \ in \ mathbb {P} \} = \ {4,7,15,21,45,75,105 \}}$

Let us consider the set S of all natural numbers n with the following property:

For every natural number k with k > 0 and 2 ^k < n , n - 2 ^{k is} a prime number .

Then S certainly contains the numbers .

{\ displaystyle 4,7,15,21,45,75,105}

For example, 45 in S because the numbers , , , , all prime numbers.

{\ displaystyle 45-2 = 43}

{\ displaystyle 45-4 = 41}

{\ displaystyle 45-8 = 37}

{\ displaystyle 45-16 = 29}

{\ displaystyle 45-32 = 13}

The conjecture now says that S only consists of these 7 numbers.

Until this assumption has been recalculated, i.e. H. there are certainly no numbers in S other than those mentioned that are less than ²⁷⁷ .

{\ displaystyle n = 2 ^ {77}}

Every number n in S (except 4) automatically yields a prime number twin , namely .

{\ displaystyle (n-2, n-4)}

See also: Follow A039669 in OEIS

Erdős divergence conjecture : It says that for every infinite sequence of the numbers +1 and −1 there are equidistant samples of finite length that add up to a sum of any magnitude. Terence Tao presented evidence in 2015.
Erdős-Woods conjecture : any whole number is given . Then there is a positive integer such that it is uniquely determined by the list of prime factors of . ${\ displaystyle n}$ ${\ displaystyle k}$ ${\ displaystyle n}$ ${\ displaystyle n, n + 1, \ dotsc, n + k}$

Conjectures on graph theory

Erdős-Faber-Lovász conjecture: A graph that is a union of complete graphs with nodes that have at most one node in common in pairs is -chromatic. ${\ displaystyle k}$ ${\ displaystyle k}$
Erdős-Gyárfás conjecture: Every graph whose vertices are all at least degree 3 contains a circle whose length is a power of two.

Assumptions about the Ramsey theory

Many conjectures, which come from Erdős or in which Erdős was involved, concern the area of the Ramsey theory and in particular the Ramsey numbers . The Bondy and Erdős conjecture and the Erdős-Sós conjecture are outstanding examples .

Web links

FRK Chung: Open problems of Paul Erdos in graph theory (PDF; 323 kB)

Individual evidence

↑ Chris Cesare: Maths whiz solves a master's riddle . Nature News, September 25, 2015.

[1] Chris Cesare: Maths whiz solves a master's riddle . Nature News, September 25, 2015.