Does it sound random? - Regards Perfect numbers - Page 2

Dr Sardonicus · 2022-05-06, 15:06

Assuming a multinomial distribution with 32 random variates X₁, ... X₃₂, each with probability 1/32, we have for 46 trials that the mean value of each X_i is $\mu\;=\; 1.4375$ , with standard deviation $\sigma\;=\; 1.18$ , approximately.

But, not knowing how the OP's distribution is defined, it is impossible to say whether it "should be" random. For all I know, the procedure might be, "Let p be prime. Define f(p) as follows: Flip a fair coin five times. Interpret Heads as 1, Tails as 0, and the result of the five flips as a number n from 0 to 31 in binary. Then f(p) = n + 1."

There are 32 odd remainders modulo 64, and 32 remainders relatively prime to 51, 68 or 80. The first case excludes one Mersenne prime (p = 2), the second excludes p = 3 and p = 17, and the other two cases each exclude p = 2, plus an additional exponent (p = 17 and p = 5, respectively).

However, the coin-flip function and the residue class of the exponent modulo 64, 51, 68, or 80 are very easy to compute. The residue class would be one of the 32 relatively prime ones for 50 of the 51 known Mersenne prime exponents modulo 64, and 49 of them modulo 51, 68 or 80.

Which brings up what to me is the most mysterious thing about the OP's question: Why was whatever the property is, checked for only 46 Mersenne prime exponents?

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2022-05-06, 15:06	#12
Dr Sardonicus Feb 2017 Nowhere 2⁶·7·13 Posts	Assuming a multinomial distribution with 32 random variates X₁, ... X₃₂, each with probability 1/32, we have for 46 trials that the mean value of each X_i is $\mu\;=\; 1.4375$ , with standard deviation $\sigma\;=\; 1.18$ , approximately. But, not knowing how the OP's distribution is defined, it is impossible to say whether it "should be" random. For all I know, the procedure might be, "Let p be prime. Define f(p) as follows: Flip a fair coin five times. Interpret Heads as 1, Tails as 0, and the result of the five flips as a number n from 0 to 31 in binary. Then f(p) = n + 1." There are 32 odd remainders modulo 64, and 32 remainders relatively prime to 51, 68 or 80. The first case excludes one Mersenne prime (p = 2), the second excludes p = 3 and p = 17, and the other two cases each exclude p = 2, plus an additional exponent (p = 17 and p = 5, respectively). However, the coin-flip function and the residue class of the exponent modulo 64, 51, 68, or 80 are very easy to compute. The residue class would be one of the 32 relatively prime ones for 50 of the 51 known Mersenne prime exponents modulo 64, and 49 of them modulo 51, 68 or 80. Which brings up what to me is the most mysterious thing about the OP's question: Why was whatever the property is, checked for only 46 Mersenne prime exponents?