every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers
Theorem 1.
If n>1540539, then n=a+b, where a and b are abundant numbers.
Proof.
Note that both 20 and 81081 are abundant numbers.
Furthermore, we have 81081=4054⋅20+1.
If n is a multiple of 20, then n-20 is also
a multiple of 20 hence, as a multiple of an abundant
number, is also abundant, so we may choose a=20
and b=n-20. Otherwise, write n=20m+k where
m and k are positive and k<20. Note that,
since n>1540539 and k<20, it follows that
m>77026>4054k, hence we have
n=20(m-4054k)+81081k. |
Since positive multiples of abundant numbers are abundant, we may set a=20(m-4054k) and b=81081k. ∎
Title | every sufficiently large even integer can be expressed as the sum of a pair of abundant numbers |
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Canonical name | EverySufficientlyLargeEvenIntegerCanBeExpressedAsTheSumOfAPairOfAbundantNumbers |
Date of creation | 2013-03-22 16:46:58 |
Last modified on | 2013-03-22 16:46:58 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 9 |
Author | rspuzio (6075) |
Entry type | Proof |
Classification | msc 11A05 |