100$ prize for the solution of a conjecture

[M344587487]

At the risk of abusing the definition, the law of small numbers or something to that effect probably applies.

[enzocreti]

Originally Posted by M344587487

At the risk of abusing the definition, the law of small numbers or something to that effect probably applies.

I dont think so...maybe the sequence is not random but i havent yet find somebody who is able to give an explanation.

[enzocreti]

Originally Posted by M344587487

At the risk of abusing the definition, the law of small numbers or something to that effect probably applies.

Here i dont see small numbers, my numbers are huge

[paulunderwood]

I think M344587487 meant a small sample size.

[enzocreti]

Originally Posted by paulunderwood

I think M344587487 meant a small sample size.

Yes but the coincidences here are too many, there is no prime congruent to 6 mod 7, five primes in a row congruent to 5 mod 7, nine primes in a row congruent to 3 mod 7. I think that if you calculate the chances that happening by pure chance are very very low

[M344587487]

Originally Posted by paulunderwood

I think M344587487 meant a small sample size.

Yes thank you. The latest sequence posted only has 34 terms.

Originally Posted by enzocreti

Yes but the coincidences here are too many, there is no prime congruent to 6 mod 7, five primes in a row congruent to 5 mod 7, nine primes in a row congruent to 3 mod 7. I think that if you calculate the chances that happening by pure chance are very very low

Million to one odds come true all the time. There are misleading patterns in everything if you massage the data enough to find them.

Time to embarrass myself with my naive maths:

Consider the odds of five in a row being congruent to c mod 7 for a fixed integer c. Assume the odds are evenly distributed (aka 1 in 7 chance across the board). The chance of finding five in a row that yield the same result mod 7 is (1/7)^4 = ~0.04164931% for every start index (with no knowledge of nearby indexes).

Now instead assume the estimated distribution to hold for the PRPs:

Originally Posted by Dr Sardonicus

The numbers in question, when viewed as strings of decimal digits, are obtained by concatenating the decimal digits of 2^k - 1 and 2^(k-1) - 1. This gives the first few terms

10, 31, 73, 157, 3115, 6331, 12763,...

Since the question concerned those of the numbers which happened to be congruent to 6 (mod 7), I computed the remainders (mod 7) out to k = one million, and found the following approximate proportions of k giving these remainders:

r = 0: 1/18
r = 1: 1/18
r = 2: 1/9
r = 3: 4/9
r = 4: 1/9
r = 5: 1/9
r = 6: 1/9

Perhaps someone might like to have a go at investigating this
;-)

Applied to five in a row congruent to 5 mod 7 we get (1/9)^4 = ~0.01524158% chance for a random start index. Applied to nine in a row congruent to 3 mod 7 we get (4/9)^8 = ~0.15224388% for a random start index. These are pretty good odds and they get much better the more indexes you allow to use as the start index.

[Dr Sardonicus]

Just to address the point of "no prp congruent to 6 mod 7," the probability of being congruent to something other than 6 (mod 7) is 1 - 1/9 or 8/9, so the probability of getting "not 6 mod 7" 34 times in a row by pure chance is

(8/9)^34 = 0.018231157+,

so it's about a 1.8% chance. That doesn't seem terribly improbable to me.

[enzocreti]

Originally Posted by Dr Sardonicus

Just to address the point of "no prp congruent to 6 mod 7," the probability of being congruent to something other than 6 (mod 7) is 1 - 1/9 or 8/9, so the probability of getting "not 6 mod 7" 34 times in a row by pure chance is

(8/9)^34 = 0.018231157+,

so it's about a 1.8% chance. That doesn't seem terribly improbable to me.

Ok but would you admit that all that coincidences at once are quite surprising? I wonder if this sequence is random

[enzocreti]

for example I calculated that the chance of five prp congruent to 5 mod 7 this early is about 1/2206. Nine primes congruent to 3 mod(7) i don't know but surely < 0.5%. Multiply all these chances and you will get an extremely low chance.

[enzocreti]

Someone told me that this sequence is not random, but he didn't give me any explanation. I searched on internet, but i didn't find any trace of that sequence. In Oeis there wasn't and so I entered it. Somebody knows if there is some hidden structure in this sequence?

[M344587487]

Originally Posted by enzocreti

Ok but would you admit that all that coincidences at once are quite surprising? I wonder if this sequence is random

The odds of a very specific thing happening may be low but you're not looking only for that specific thing. It's a trap to think like that. The odds of my exact genetic code being produced through hundreds of generations and my subsequent development leading to being interested in primes enough to be on this forum and talking to you are so vanishingly small it's incredible. The odds of someone reading your comment and taking a similar stance as me if I weren't here are high.

Originally Posted by enzocreti

Someone told me that this sequence is not random, but he didn't give me any explanation. I searched on internet, but i didn't find any trace of that sequence. In Oeis there wasn't and so I entered it. Somebody knows if there is some hidden structure in this sequence?

You're essentially looking for the intersection between the set of prime numbers and the set of numbers given by an equation with no randomness. Prime numbers are not random and the equation is not random, so the intersection cannot be random. It's just hideously complex, far more complex than we can unravel. We can tease out some properties based on what we know about the two sets, but we can't predict the next term in the sequence with any degree of accuracy any more than we can predict the next Mersenne prime.