Interesting post (as always). There is something ever so slightly funky to me that perfect numbers include 1 since 1 seems often ignored as a factor. But the alternate definition, the prime being half the sum of *all* its factors, doesn't seem as "perfect".
The natural way to define perfect numbers is in terms of the “sum of divisors” function, σ, which defined to be the sum of *all* the factors, including the number itself. A number n is then perfect if σ(n)=2n. The function σ is a good example of a multiplicative (or “arithmetic”) function, which means that σ(mn)= σ(m)σ(n) whenever m and n are coprime. It follows that σ(1)=1, so that 1 is *not* a perfect number. If we tried to do something similar with the “sum of all divisors apart from the number itself” function, this wouldn’t work.
Another way to look at this is that if it could be proved that there were infinitely many Goormaghtigh numbers, then it would follow that the abc conjecture was false, and this would be a major result.
Interesting post (as always). There is something ever so slightly funky to me that perfect numbers include 1 since 1 seems often ignored as a factor. But the alternate definition, the prime being half the sum of *all* its factors, doesn't seem as "perfect".
The natural way to define perfect numbers is in terms of the “sum of divisors” function, σ, which defined to be the sum of *all* the factors, including the number itself. A number n is then perfect if σ(n)=2n. The function σ is a good example of a multiplicative (or “arithmetic”) function, which means that σ(mn)= σ(m)σ(n) whenever m and n are coprime. It follows that σ(1)=1, so that 1 is *not* a perfect number. If we tried to do something similar with the “sum of all divisors apart from the number itself” function, this wouldn’t work.
Makes sense, thanks for the explanation.
Another way to look at this is that if it could be proved that there were infinitely many Goormaghtigh numbers, then it would follow that the abc conjecture was false, and this would be a major result.