Editing Talk:410: Math Paper
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Despite what this comic implies, the divisor function is defined over the Gaussian integers. There still is a problem, though. If a divides b, then so does -a, along with ai and -ai. The divisors will inevitably sum to zero. You could get around this by ignoring all the numbers that aren't in a given quadrant. I personally like the idea of using ones where the real part is greater than the imaginary part (although that still does become a problem with multiples of 1+i). This way, a friend of a natural number will also be a natural number (though it's only the same as what you'd get normally if all the factors are three mod four). {{unsigned ip|199.27.128.167}} | Despite what this comic implies, the divisor function is defined over the Gaussian integers. There still is a problem, though. If a divides b, then so does -a, along with ai and -ai. The divisors will inevitably sum to zero. You could get around this by ignoring all the numbers that aren't in a given quadrant. I personally like the idea of using ones where the real part is greater than the imaginary part (although that still does become a problem with multiples of 1+i). This way, a friend of a natural number will also be a natural number (though it's only the same as what you'd get normally if all the factors are three mod four). {{unsigned ip|199.27.128.167}} | ||
| β | :I do not agree. Here is how it should work. You should define that a divides b if and only if there is a natural number n such that an = b. This way, natural numbers don't get new divisors when you move to the Gaussian plane. Consequently the extended sigma function gives the same value as the classical one when applied to a natural number. So, natural numbers will be friends according to the new definition if and only if they are friends according to the old definition and we are indeed allowed to say that the new definition extends the old one (Burghard von Karger). | + | :I do not agree. Here is how it should work. You should define that a divides b if and only if there is a natural number n such that an = b. This way, natural numbers don't get new divisors when you move to the Gaussian plane. Consequently the extended sigma function gives the same value as the classical one when applied to a natural number. So, natural numbers will be friends according to the new definition if and only if they are friends according to the old definition and we are indeed allowed to say that the new definition extends the old one (Burghard von Karger). |
:So you're saying that the only friends imaginary friends have are imaginary? [[User:Firestar233|guess who]] ([[User talk:Firestar233|if you want to]] | [[Special:Contributions/Firestar233|what i have done]]) 23:21, 2 November 2023 (UTC) | :So you're saying that the only friends imaginary friends have are imaginary? [[User:Firestar233|guess who]] ([[User talk:Firestar233|if you want to]] | [[Special:Contributions/Firestar233|what i have done]]) 23:21, 2 November 2023 (UTC) | ||
