Abc conjecture

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“The abc conjecture is the most important unsolved problem in diophantine analysis.”—Dorian Goldfeld

Joseph Oesterlé (1988) and David Masser (1985) proposed the following conjecture:

Conjecture (ABC conjecture, Oesterlé–Masser conjecture, 1985 and 1988). (Oesterlé and Masser)

For any

ϵ > 0

, there is a constant

μϵ > 1

such that if

a

and

b

are coprime [negative or positive] integers and

c = a + b

, then

max (  | a | , | b | , | c |  )  ≤  μϵ (N (a, b, c)) 1 + ϵ, a, b, c ∈ ℤ∖{0},

where

N (a, b, c)   :=   rad (a b c)

is the squarefree kernel, or radical (i.e. product of distinct prime factors) of

a b c

.

The conjecture may be restated as

max (  | a | , | b | , | c |  )  ≤  μϵ (rad (a, b, c)) 1 + ϵ  =  μϵ     p∣a b c ∏   p∣a b c    p  1 + ϵ, a , b, c ∈ ℤ∖{0},

or

max (  | a | , | b | , | c |  )  ≤  μϵ (rad (a) rad (b) rad (c)) 1 + ϵ  =  μϵ{    p∣a ∏   p∣a    p    p∣b ∏   p∣b    p    p∣c ∏   p∣c    p}  1 + ϵ, a , b, c ∈ ℤ∖{0}.

The above is the same as considering

a, b

and

c

to be all positive coprime integers, after swapping (across the = sign) and renaming the variables such that

c = a + b

, thus we have

c  ≤  μϵ (N (a, b, c)) 1 + ϵ, a, b, c ∈ ℕ +.

Although the conjecture is well-established, there were no obvious strategies for resolving the problem. It is still unsolved.

Abc conjecture as hypothesis

The truth of the abc conjecture would have consequences for

• Fermat’s last theorem (a much simpler proof of...) See the answer by quid on MathOverflow.
• Beal’s conjecture (which would then hold when the exponents are large enough)
• Catalan’s conjecture (Mihăilescu’s theorem)
• ...

Sequences

A085152 Sequence related to ABC conjecture: All prime factors of

n

and

n + 1

are

≤   5

.

{1, 2, 3, 4, 5, 8, 9, 15, 24, 80, ...}

A085153 Sequence related to ABC conjecture: all prime factors of

n

and

n + 1

are

≤   7

.

{1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, ...}

A051037 5-smooth numbers: i.e. numbers whose prime divisors are all

≤   5

.

{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, ...}

A002473 Highly composite numbers (2): numbers whose prime divisors are all

≤   7

.

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, ...}

See also

• p-smooth numbers
• x y z conjecture

External links

• Dorian Goldfeld, Modular Forms, Elliptic Curves and the ABC–Conjecture.
• Dorian Goldfeld, Beyond the Last Theorem, The Sciences, Published by the New York Academy of Sciences, March/April 1996, pp. 34–40.
• Andrew Granville and Thomas J. Tucker, It’s As Easy As a b c, Notices of the AMS, Nov 2002.
• Weisstein, Eric W., abc Conjecture, from MathWorld—A Wolfram Web Resource.
• abc conjecture—Wikipedia.org.
• Ivars Peterson’s MathTrek, The Amazing ABC Conjecture.
• Paulo Ribenboim’s letter to A. Nitaj, The a b c conjecture is as hard as the x y z conjecture, 19 August, 2002.