[math-fun] Question about "central" binomial coefficients binomial(2n+1, n+1)
The OEIS sequence for binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!) (<https://oeis.org/A001700>) begins with 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052, (0 ≤ n ≤ 25), for which binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1. Is this true for all n ≥ 0 ??? —Dan
In the following pattern, zeroes mark odds, other integers count factors of two, arranged in triangular array to show binary structure. To obtain non-zero values on each line, copy sequence P from line above then append 1 followed by P+1. 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 0, and the next line is: 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 0. This should all be provable from the recurrence relation: a(n) = 2*(2*n+1)*a(n-1)/(n+1) when you take the denominator as (n+1)=2^k, but I didn't bother to work out any more details, sorry. This type of structure is reminiscent of Cellular Automaton Rule 90, see also: https://demonstrations.wolfram.com/Code686BuildsTheChairTiling/ This reminds me that the parity pattern of pascal's triangle is also code 90, so you could probably prove it that way. --Brad On Tue, Jan 26, 2021 at 6:12 PM Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract. On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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And this works for any prime, not just 2. On Tue, Jan 26, 2021 at 5:33 PM Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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Thanks! I wonder if there's a more concrete way to describe the power of 2. I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day. (I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.) —Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The observation of Kummer eventually lead to the p-aficionado Gamma function of Morris, the https://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula?wprov=sfti1 And the theory of p-ADVI differential equations of Dwork. On Tue, Jan 26, 2021 at 20:46 Dan Asimov <asimov@msri.org> wrote:
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller < victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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It should be p-adic differential equations and the Gross Koblitz formula On Tue, Jan 26, 2021 at 20:55 Victor Miller <victorsmiller@gmail.com> wrote:
The observation of Kummer eventually lead to the p-aficionado Gamma function of Morris, the https://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula?wprov=sfti1
And the theory of p-ADVI differential equations of Dwork.
On Tue, Jan 26, 2021 at 20:46 Dan Asimov <asimov@msri.org> wrote:
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
1.
Is this true for all n ≥ 0 ???
—Dan
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On Tuesday/26January/2021, at 5:32 PM, Victor Miller < victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Autocorrect messed this up. It should be the p-adic Gamma function of Morita On Tue, Jan 26, 2021 at 20:56 Victor Miller <victorsmiller@gmail.com> wrote:
It should be p-adic differential equations and the Gross Koblitz formula
On Tue, Jan 26, 2021 at 20:55 Victor Miller <victorsmiller@gmail.com> wrote:
The observation of Kummer eventually lead to the p-aficionado Gamma function of Morris, the https://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula?wprov=sfti1
And the theory of p-ADVI differential equations of Dwork.
On Tue, Jan 26, 2021 at 20:46 Dan Asimov <asimov@msri.org> wrote:
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller < victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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Dan, your method of subtraction is known in the literature as the “Austrian” method, one of the 3 main ways of subtracting in America in the 1930s. The history of why we ended up subtracting using reallocation is that forever, subtraction “using crutches” or putting markings on the paper was considered inappropriate. Then a single county administrator decided to experiment, with some of his school using crutches, some not. Those using crutches understood as well as those who didn’t what was going on, and got the answers right substantially more often. Almost overnight, a\American schools used crutches, and still to today. By sheer chance, that county used reallocation. So we all do now. I have a history paper on 10 different ways to subtract through the ages, must get that submitted at some point... Steve -- Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA<x-apple-data-detectors://1/0> Phone 540 568 5104<tel:540%20568%205104>, Fax 540 568 6857<tel:540%20568%C2%A06857>, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other) Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato) On Jan 26, 2021, at 8:46 PM, Dan Asimov <asimov@msri.org> wrote: CAUTION: This email originated from outside of JMU. Do not click links or open attachments unless you recognize the sender and know the content is safe. ________________________________ Thanks! I wonder if there's a more concrete way to describe the power of 2. I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day. (I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.) —Dan On Tuesday/26January/2021, at 5:32 PM, Victor Miller <victorsmiller@gmail.com> wrote: This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract. On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote: The OEIS sequence for binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!) (<https://urldefense.proofpoint.com/v2/url?u=https-3A__oeis.org_A001700&d=DwIG... >) begins with 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052, (0 ≤ n ≤ 25), for which binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1. Is this true for all n ≥ 0 ??? —Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...
Very interesting. (Since my method was essentially the same as doing an addition, it seemed much simpler and straightforward to me. And other than the answer it never required any markings on the page.) —Dan
On Tuesday/26January/2021, at 6:09 PM, Lucas, Stephen K - lucassk <lucassk@jmu.edu> wrote:
Dan, your method of subtraction is known in the literature as the “Austrian” method, one of the 3 main ways of subtracting in America in the 1930s. The history of why we ended up subtracting using reallocation is that forever, subtraction “using crutches” or putting markings on the paper was considered inappropriate. Then a single county administrator decided to experiment, with some of his school using crutches, some not. Those using crutches understood as well as those who didn’t what was going on, and got the answers right substantially more often. Almost overnight, a\American schools used crutches, and still to today. By sheer chance, that county used reallocation. So we all do now.
I have a history paper on 10 different ways to subtract through the ages, must get that submitted at some point...
Steve
--
Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA<x-apple-data-detectors://1/0 <x-apple-data-detectors://1/0>> Phone 540 568 5104<tel:540%20568%205104 <tel:540%20568%205104>>, Fax 540 568 6857<tel:540%20568%C2%A06857 <tel:540%20568%C2%A06857>>, Web http://educ.jmu.edu/~lucassk/ <http://educ.jmu.edu/~lucassk/> Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other)
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato)
On Jan 26, 2021, at 8:46 PM, Dan Asimov <asimov@msri.org <mailto:asimov@msri.org>> wrote:
CAUTION: This email originated from outside of JMU. Do not click links or open attachments unless you recognize the sender and know the content is safe. ________________________________
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://urldefense.proofpoint.com/v2/url?u=https-3A__oeis.org_A001700&d=DwIG... <https://urldefense.proofpoint.com/v2/url?u=https-3A__oeis.org_A001700&d=DwIGaQ&c=eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m=ByWOpi8oPdavPu9Fj9QM35o7KamteGHa7oRcRkrMzPM&s=df0CMDm6bwHnDg48nvOl-rb8C9X5JUZyaI4Ra9Mr2IA&e=> >) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... <https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cgi-2Dbin_mailman_listinfo_math-2Dfun&d=DwIGaQ&c=eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m=ByWOpi8oPdavPu9Fj9QM35o7KamteGHa7oRcRkrMzPM&s=ZCxT94iGY7AUp-n5aaa6uc-AyT7y1Hy-zaN5iEVUoWQ&e=>
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... <https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cgi-2Dbin_mailman_listinfo_math-2Dfun&d=DwIGaQ&c=eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m=ByWOpi8oPdavPu9Fj9QM35o7KamteGHa7oRcRkrMzPM&s=ZCxT94iGY7AUp-n5aaa6uc-AyT7y1Hy-zaN5iEVUoWQ&e=>
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg... <https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cgi-2Dbin_mailman_listinfo_math-2Dfun&d=DwIGaQ&c=eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m=ByWOpi8oPdavPu9Fj9QM35o7KamteGHa7oRcRkrMzPM&s=ZCxT94iGY7AUp-n5aaa6uc-AyT7y1Hy-zaN5iEVUoWQ&e=> _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <mailto:math-fun@mailman.xmission.com> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun <https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
My mother taught me to subtract with the algorithm she learned, long before the inferior algorithm I was taught in school. Borrowing from the minuend requires lots of memory; carrying into the subtrahend does not. On 26-Jan-21 21:09, Lucas, Stephen K - lucassk wrote:
Dan, your method of subtraction is known in the literature as the “Austrian” method, one of the 3 main ways of subtracting in America in the 1930s. The history of why we ended up subtracting using reallocation is that forever, subtraction “using crutches” or putting markings on the paper was considered inappropriate. Then a single county administrator decided to experiment, with some of his school using crutches, some not. Those using crutches understood as well as those who didn’t what was going on, and got the answers right substantially more often. Almost overnight, a\American schools used crutches, and still to today. By sheer chance, that county used reallocation. So we all do now.
I have a history paper on 10 different ways to subtract through the ages, must get that submitted at some point...
Steve
--
Stephen Lucas, Professor Department of Mathematics and Statistics MSC 1911, James Madison University, Harrisonburg, VA 22807 USA<x-apple-data-detectors://1/0> Phone 540 568 5104<tel:540%20568%205104>, Fax 540 568 6857<tel:540%20568%C2%A06857>, Web http://educ.jmu.edu/~lucassk/ Email lucassk at jmu dot edu (Work) stephen.k.lucas at gmail dot com (Other)
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato)
On Jan 26, 2021, at 8:46 PM, Dan Asimov <asimov@msri.org> wrote:
CAUTION: This email originated from outside of JMU. Do not click links or open attachments unless you recognize the sender and know the content is safe. ________________________________
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
On Tuesday/26January/2021, at 5:32 PM, Victor Miller <victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://urldefense.proofpoint.com/v2/url?u=https-3A__oeis.org_A001700&d=DwIG... >) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K - 1.
Is this true for all n ≥ 0 ???
—Dan
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I would like to see that!
On Jan 26, 2021, at 7:09 PM, Lucas, Stephen K - lucassk <lucassk@jmu.edu> wrote:
I have a history paper on 10 different ways to subtract through the ages, must get that submitted at some point...
Cris Moore moore@santafe.edu
participants (7)
-
Brad Klee -
Cris Moore -
Dan Asimov -
Lucas, Stephen K - lucassk -
Mike Speciner -
Tomas Rokicki -
Victor Miller