I agree that you can be 99.99% (or more) certain that 53 is prime but I don’t think you can be that confident based only on the arguement you gave.
If a number is composite, it must have a prime factor no greater than its square root. Because 53 is less than 64, sqrt(53) is less than 8. So, to find out if 53 is prime or not, we only need to check if it can be divided by primes less than 8 (i.e. 2, 3, 5, and 7). 53′s last digit is odd, so it’s not divisible by 2. 53′s last digit is neither 0 nor 5, so it’s not divisible by 5. The nearest multiples of 3 are 51 (=17x3) and 54, so 53 is not divisible by 3. The nearest multiples of 7 are 49 (=7^2) and 56, so 53 is not divisible by 7. Therefore, 53 is prime.
There are just too many potential errors that could occur in this chain of reasoning. For example, how sure are you that you correctly listed the primes less than 8? Even a mere typo at this stage of the argument could result in an erroneous conclusion.
Anyway just to be clear I do think your high confidence that 53 is prime is justified, but that the argument you gave for it is insufficient in isolation.
I wouldn’t call that argument my only reason, but it’s my best shot at expressing my main reason in words.
Funny story: when I was typing this post, I almost typed, “If a number is not prime, it must have a prime factor greater than its square root.” But that’s wrong, counterexamples include pi, i, and integers less than 2. Not that I was confused about that, my real reasoning was partly nonverbal and included things like “I’m restricting myself to the domain of integers greater than 1″ as unstated assumptions. And I didn’t actually have to spell out for myself the reasoning why 2 and 5 aren’t factors of 53; that’s the sort of thing I’m used to just seeing at a glance.
This left me fearing that someone would point out some other minor error in the argument in spite of the arguments’ being essentially correct, and I’d have to respond, “Well, I said I was 99.99% sure 53 was prime, I never claimed to be 99.99% sure of that particular argument.”
I agree that you can be 99.99% (or more) certain that 53 is prime but I don’t think you can be that confident based only on the arguement you gave.
There are just too many potential errors that could occur in this chain of reasoning. For example, how sure are you that you correctly listed the primes less than 8? Even a mere typo at this stage of the argument could result in an erroneous conclusion.
Anyway just to be clear I do think your high confidence that 53 is prime is justified, but that the argument you gave for it is insufficient in isolation.
I wouldn’t call that argument my only reason, but it’s my best shot at expressing my main reason in words.
Funny story: when I was typing this post, I almost typed, “If a number is not prime, it must have a prime factor greater than its square root.” But that’s wrong, counterexamples include pi, i, and integers less than 2. Not that I was confused about that, my real reasoning was partly nonverbal and included things like “I’m restricting myself to the domain of integers greater than 1″ as unstated assumptions. And I didn’t actually have to spell out for myself the reasoning why 2 and 5 aren’t factors of 53; that’s the sort of thing I’m used to just seeing at a glance.
This left me fearing that someone would point out some other minor error in the argument in spite of the arguments’ being essentially correct, and I’d have to respond, “Well, I said I was 99.99% sure 53 was prime, I never claimed to be 99.99% sure of that particular argument.”