[math-fun] primes in geometric progressionm new record.
Hello, I continued the calculation of large primes with the function { c^k } , and found that if c = 999982.6807... then, 899 primes are generated. http://plouffe.fr/NEW/899%20primes%20in%20geometric%20progression.txt By the way, I compared the speed of calculation for <isprime> in Maple, Mathematica, Pari-Gp and PFGW. PFGW is the fastest for testing if a prime is probable, well at least for large values. But if you want from a given n to find the next prime after n then Maple is the fastest. I could not use PFGW for doing that at all, too slow. The function nextprime() within PFGW is <mickey mouse> , it works for small values but fails miserably for n > 10^20. I tried an intelligent loop on it and it is by far too slow. I believe that for once, Maple is the best. Which brings up the next question. Emile Borel once calculated what could be what is considered impossible on a human, terrestrial or cosmic scale. If I remember correctly, a probability of 1/1000000 is on a human scale impossible. It was 10 ^ (- 12) for the terrestrial scale and 10 ^ (- 50) for the cosmic scale. In other words, if you calculate that there is less than 1 in 1 million chance that an unfortunate event can happen to you, then you can rest easy and if it is on a cosmic scale then all of humanity can also sleep peacefully. Which brings me to the next question. Does anyone know what is the probability that a probable prime number of 1000 digits can in fact be composed like a Carmichael number? Best regards (and chance) to you, Simon Plouffe
I don't have my Knuth books with me at the moment, but I remember a probabilistic primality tests that relied on a choice of random numbers, and that the probability of a false positive was 25%. Since the tests were independent, he argued it would be extremely unlikely to find a number that is composite yet passes this test e.g. 25 times in a row. The test is described here: https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test One could then try feeding this algorithm composites of forms such as the Carmichael numbers and see how the test behaves. Andres. On 6/15/20 21:54, Simon Plouffe wrote:
Hello, I continued the calculation of large primes with the function { c^k } , and found that if c = 999982.6807... then, 899 primes are generated. http://plouffe.fr/NEW/899%20primes%20in%20geometric%20progression.txt
By the way, I compared the speed of calculation for <isprime> in Maple, Mathematica, Pari-Gp and PFGW. PFGW is the fastest for testing if a prime is probable, well at least for large values.
But if you want from a given n to find the next prime after n then Maple is the fastest. I could not use PFGW for doing that at all, too slow. The function nextprime() within PFGW is <mickey mouse> , it works for small values but fails miserably for n > 10^20. I tried an intelligent loop on it and it is by far too slow. I believe that for once, Maple is the best.
Which brings up the next question. Emile Borel once calculated what could be what is considered impossible on a human, terrestrial or cosmic scale. If I remember correctly, a probability of 1/1000000 is on a human scale impossible. It was 10 ^ (- 12) for the terrestrial scale and 10 ^ (- 50) for the cosmic scale. In other words, if you calculate that there is less than 1 in 1 million chance that an unfortunate event can happen to you, then you can rest easy and if it is on a cosmic scale then all of humanity can also sleep peacefully. Which brings me to the next question.
Does anyone know what is the probability that a probable prime number of 1000 digits can in fact be composed like a Carmichael number?
Best regards (and chance) to you, Simon Plouffe _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
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Andres Valloud -
Simon Plouffe