every even integer greater than 70 is the sum of two abundant numbers in more than one way
To prove this it is enough to find just two ways for each even , though of course there are plenty more ways as the numbers get larger, purely for our convenience we’ll seek to choose the smallest values for possible. Since every multiple of a perfect number is an abundant number, and 6 is a perfect number, it follows that every multiple of 6 is abundant, and it is small enough a modulus that reviewing all possible cases should not prove tiresome.
If and , the two desired pairs are , , and , . This leaves us the cases and to concern ourselves with.
If and then the pairs are are , , and , .
If and then the pairs are , , and , .
The lower bounds of have been chosen to ensure the formulas give distinct pairs of abundant numbers and never the perfect number 6 itself, but its multiples. These values of correspond to the values of 36, 68, 82. To complete the proof we are left with the special case of to examine on its own. Ignoring the bounds for , the formulas above give us 76 = 40 + 36, a valid pair, and 76 = 70 + 6, which is not a pair of abundant numbers. But there is one other pair, 56 + 20, of which neither nor is a multiple of 6. ∎
The special case of 76 shows that there are solutions that don’t use multiples of 6. These become more readily available as the numbers get larger.
|Title||every even integer greater than 70 is the sum of two abundant numbers in more than one way|
|Date of creation||2013-03-22 17:44:14|
|Last modified on||2013-03-22 17:44:14|
|Last modified by||PrimeFan (13766)|