[math-fun] period of fibonacci sequence mod n
Obviously, the greatest period we could hope for would be (n*n-1). This is achieved when n=2 and n=3, but for no other n up to 270000. If we merely ask that period>=(n*n-1)/2, or period>=n^(3/2), that happens for n=2,3,5,6,10, but for no other n up to 270000. If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n. Periods exceeding n seem to happen about 10% of the time. No case where period>10*n happens up to 270000. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n.
--that's funny. Rerunning my program, this time it said n=156250 obeyed the rule (?hardware bug?), and indeed it claims the rule works for each n=2*5^k for k=1,2,3,4,5,6,7,8,9,10,11.
See https://oeis.org/A001175, one of the oldest entries in the OEIS. If your observations aren't mentioned there, please add them! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, Jan 24, 2016 at 12:10 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n.
--that's funny. Rerunning my program, this time it said n=156250 obeyed the rule (?hardware bug?), and indeed it claims the rule works for each n=2*5^k for k=1,2,3,4,5,6,7,8,9,10,11.
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Since Warren was interested in extreme values for A001175(n)/n, I went looking for two obvious associated sequences: indices at which A001175 hits a new record (1, 2, 3, 5, 6, 10, 25 ...) and the corresponding record values (1, 3, 8, 20, 24 ...) and neither were present. I have given here the minimum number of terms required to produce an OEIS miss. The biggest ratio I can see by eye is A001175(6250)/6250 = 6. On Mon, Jan 25, 2016 at 12:28 AM, Neil Sloane <njasloane@gmail.com> wrote:
See https://oeis.org/A001175, one of the oldest entries in the OEIS.
If your observations aren't mentioned there, please add them!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sun, Jan 24, 2016 at 12:10 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n.
--that's funny. Rerunning my program, this time it said n=156250 obeyed the rule (?hardware bug?), and indeed it claims the rule works for each n=2*5^k for k=1,2,3,4,5,6,7,8,9,10,11.
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Alan, Please submit them right away! One of the gang will then surely extend them. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Jan 25, 2016 at 6:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Since Warren was interested in extreme values for A001175(n)/n, I went looking for two obvious associated sequences: indices at which A001175 hits a new record (1, 2, 3, 5, 6, 10, 25 ...) and the corresponding record values (1, 3, 8, 20, 24 ...) and neither were present. I have given here the minimum number of terms required to produce an OEIS miss.
The biggest ratio I can see by eye is A001175(6250)/6250 = 6.
On Mon, Jan 25, 2016 at 12:28 AM, Neil Sloane <njasloane@gmail.com> wrote:
See https://oeis.org/A001175, one of the oldest entries in the OEIS.
If your observations aren't mentioned there, please add them!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sun, Jan 24, 2016 at 12:10 PM, Warren D Smith <warren.wds@gmail.com> wrote:
If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n.
--that's funny. Rerunning my program, this time it said n=156250 obeyed the rule (?hardware bug?), and indeed it claims the rule works for each n=2*5^k for k=1,2,3,4,5,6,7,8,9,10,11.
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Will try to remember to do this tonight. On Mon, Jan 25, 2016 at 7:07 PM, Neil Sloane <njasloane@gmail.com> wrote:
Alan, Please submit them right away! One of the gang will then surely extend them.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Mon, Jan 25, 2016 at 6:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Since Warren was interested in extreme values for A001175(n)/n, I went looking for two obvious associated sequences: indices at which A001175 hits a new record (1, 2, 3, 5, 6, 10, 25 ...) and the corresponding record values (1, 3, 8, 20, 24 ...) and neither were present. I have given here the minimum number of terms required to produce an OEIS miss.
The biggest ratio I can see by eye is A001175(6250)/6250 = 6.
On Mon, Jan 25, 2016 at 12:28 AM, Neil Sloane <njasloane@gmail.com> wrote:
See https://oeis.org/A001175, one of the oldest entries in the OEIS.
If your observations aren't mentioned there, please add them!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sun, Jan 24, 2016 at 12:10 PM, Warren D Smith <warren.wds@gmail.com
wrote:
If we merely ask that period>=5*n, that happens for n=10,50,250,1250,6250,31250, and which all are of form 2*5^k, but for no other n up to 270000. Note in particular that n=156250=2*5^7 is NOT listed. In all six of these cases, the period equals 6*n.
--that's funny. Rerunning my program, this time it said n=156250 obeyed the rule (?hardware bug?), and indeed it claims the rule works for each n=2*5^k for k=1,2,3,4,5,6,7,8,9,10,11.
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participants (3)
-
Allan Wechsler -
Neil Sloane -
Warren D Smith