[math-fun] Favourite prime numbers
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. ))) But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers? --Dan Sometimes the brain has a mind of its own.
My favorite prime is 144169. Besides the fact that it looks like 12^2 concatenated with 13^2, it's also the discriminant of the quadratic field which contains the eigenvalues of the Hecke operators on modular forms of weight 24. Victor On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. )))
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers?
--Dan
Sometimes the brain has a mind of its own.
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149459, a prime number, the period of the octal game 0.16 It magically appeared in the green phosphor of a DEC VT100 in the Stanford CS department basement in the fall of 1986 On Fri, Jun 3, 2011 at 6:26 AM, Victor Miller <victorsmiller@gmail.com> wrote:
My favorite prime is 144169. Besides the fact that it looks like 12^2 concatenated with 13^2, it's also the discriminant of the quadratic field which contains the eigenvalues of the Hecke operators on modular forms of weight 24.
Victor
On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. )))
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers?
--Dan
Sometimes the brain has a mind of its own.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Let's not forget 314159! --Rich ----- Quoting Thane Plambeck <tplambeck@gmail.com>:
149459, a prime number, the period of the octal game 0.16
It magically appeared in the green phosphor of a DEC VT100 in the Stanford CS department basement in the fall of 1986
On Fri, Jun 3, 2011 at 6:26 AM, Victor Miller <victorsmiller@gmail.com> wrote:
My favorite prime is 144169. Besides the fact that it looks like 12^2 concatenated with 13^2, it's also the discriminant of the quadratic field which contains the eigenvalues of the Hecke operators on modular forms of weight 24.
Victor
On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. )))
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers?
--Dan
Sometimes the brain has a mind of its own.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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My favorite prime is 1+i. On Fri, Jun 3, 2011 at 3:01 PM, <rcs@xmission.com> wrote:
Let's not forget 314159! --Rich
----- Quoting Thane Plambeck <tplambeck@gmail.com>:
149459, a prime number, the period of the octal game 0.16
It magically appeared in the green phosphor of a DEC VT100 in the Stanford CS department basement in the fall of 1986
On Fri, Jun 3, 2011 at 6:26 AM, Victor Miller <victorsmiller@gmail.com> wrote:
My favorite prime is 144169. Besides the fact that it looks like 12^2 concatenated with 13^2, it's also the discriminant of the quadratic field which contains the eigenvalues of the Hecke operators on modular forms of weight 24.
Victor
On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. )))
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers?
--Dan
Sometimes the brain has a mind of its own.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Another interesting prime: 43,252,003,274,489,855,999 The number of unsolved positions of Rubik's cube. On Fri, Jun 3, 2011 at 3:05 PM, Mike Stay <metaweta@gmail.com> wrote:
My favorite prime is 1+i.
On Fri, Jun 3, 2011 at 3:01 PM, <rcs@xmission.com> wrote:
Let's not forget 314159! --Rich
----- Quoting Thane Plambeck <tplambeck@gmail.com>:
149459, a prime number, the period of the octal game 0.16
It magically appeared in the green phosphor of a DEC VT100 in the Stanford CS department basement in the fall of 1986
On Fri, Jun 3, 2011 at 6:26 AM, Victor Miller <victorsmiller@gmail.com> wrote:
My favorite prime is 144169. Besides the fact that it looks like 12^2 concatenated with 13^2, it's also the discriminant of the quadratic field which contains the eigenvalues of the Hecke operators on modular forms of weight 24.
Victor
On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov@earthlink.net> wrote:
((( My favorite number is 24 hands down, because of its amazing properties like showing up in Dedekind eta, 1^2 + ... + 24^2 = 70^2 being the only such relation, the 24-dimensional Leech lattice, and 24 being the largest number N such that all smaller numbers K with GCD(K,N) = 1 satisfy K^2 == 1 (mod N). At least some of these are interconnected. )))
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ? Are there standard measures of some sort that distinguish among prime numbers?
--Dan
Sometimes the brain has a mind of its own.
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
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="Dan Asimov" <dasimov@earthlink.net> My favorite number is 24 hands down, because of its amazing properties like showing up in [...]
It's interesting how diverse people's esthetics can be. I find 24 pleasant for all the reasons cited, but feel this ubiquity and regularity make it a bit too stolid and boring. Perhaps I'm too inclined to novelty, but when an answer turns out to be a number like 24 it's like learning that breakfast is a bowl of cornflakes and skim milk. In contrast, I tend to prefer eruptions of unexpected unique identity against an apparently uniform background--like a frog leaping out of a bowl of curry! So my "canonical" favorite number is 691.
But how about prime numbers? After the first few, how do you distinguish, say, 101, 103, 107, 109 ?
I like Ramanujan's famous exhortation to "make friends with the integers", and tend to view unfamiliar numbers as friends I haven't yet made, but can look forward to getting to know better, should the opportunity arise. I find it particularly ironic that this list ends with 109, which lately to me is an attention-grabbing SETI signal that's associated with some topics of personal interest! See for example the thread on math-fun from last November with subject "Coincidentally 109", and OEIS sequence A054244 and others related to (binary) "dismal arithmetic".
Are there standard measures of some sort that distinguish among prime numbers?
That's an interesting idea. I prefer primes that aren't tweaks of vanilla "round" numbers. By that standard I find 17 to be much less cool than 19. It's not at all clear, though, how to quantify that. Intuitively, there might be some kind of complexity measure, such as the simplest/shortest recurrence that generates N (say using just 1, addition and functional composition)?
My favorite primes are 91, 2 (the Odd prime) and 8 (largest even prime). I'm also a big fan of the Fermat Half-Primes: 4,6,15,91,703,1891,2701,... Citations: [91 and 2]: http://fortunes.cat-v.org/openbsd/ (search for "The Odd Prime") [8]: http://xkcd.com/899/ [Fermat Half-Prime]: n such that 2|a|=|b|, for a in 1<a<n, a^(n-1)%n == 1, and 0<b<n relatively prime to n.
[Fermat Half-Prime]: n such that 2|a|=|b|, for a in 1<a<n, a^(n-1)%n == 1, and 0<b<n relatively prime to n.
(Bug: it should be 0<a<n above.) Actually, this is somewhat interesting. The "Fermat Half-Primes" in 2..100000 are: 4,6,15,91,703,1891,2701,11305,12403,13981,18721,23001,30889,38503,39865,49141,68101,79003,88561,88831,91001,93961. OEIS lists has "A129521: Numbers of the form p*q, p and q prime with q=2*p-1", which appears to be a subset of this sequence: 6,15,91,703,1891,2701,12403,18721,38503,49141,79003,88831,104653,... (The Half-Primes not in A129521 are: 4,11305,13981,23001,30889,39865,68101,88561,91001,93961).
On Sunday 05 June 2011 02:19:58 Jason wrote:
[Fermat Half-Prime]: n such that 2|a|=|b|, for a in 1<a<n, a^(n-1)%n == 1, and 0<b<n relatively prime to n.
(Bug: it should be 0<a<n above.)
Actually, this is somewhat interesting. The "Fermat Half-Primes" in 2..100000 are:
4,6,15,91,703,1891,2701,11305,12403,13981,18721,23001,30889,38503,39865,491 41,68101,79003,88561,88831,91001,93961.
OEIS lists has "A129521: Numbers of the form p*q, p and q prime with q=2*p-1", which appears to be a subset of this sequence:
6,15,91,703,1891,2701,12403,18721,38503,49141,79003,88831,104653,...
Yup. When n=pq with p,q=2p-1 prime, a^(n-1) = 1 (mod p) iff a is a quadratic residue mod q: thus, half the time. Incidentally, I think your definition of "Fermat half-prime" would have been clearer if more explicit: "n such that exactly half of the a such that 0<a<n and (a,n)=1 satisfy a^(n-1) = 1 (mod p)". -- g
participants (9)
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Dan Asimov -
Gareth McCaughan -
Jason -
Marc LeBrun -
Mike Stay -
rcs＠xmission.com -
Thane Plambeck -
Tom Rokicki -
Victor Miller