sinclair@aero.ARPA (William S. Sinclair) (08/22/85)
Most of you have probably heard the story of Ramanujan, who was riding
in the cab with a friend. They were discussing his room number 1729,
when his friend remarked that it was an uninteresting number.
"Oh no" Ramanujan replied. "it is the smallest number that can be written
as the sum of two cubes in two different ways".
My question is, what is the smallest number that can be written as the
sum of two cubes in THREE different ways? Does one exist?
Bill Sinclair (asbestos Willie)matt@oddjob.UUCP (08/26/85)
>My question is, what is the smallest number that can be written as the >sum of two cubes in THREE different ways? Does one exist? > > Bill Sinclair (asbestos Willie) The smallest 3-way sum of cubes is 87539319. Misspending part of a summer on this sort of question led me to observe that the number 221*5^n seems to be expressible as a sum of two squares 2n+2 different ways for n through at least 10 or so. Is it true for all n? If so, why? _____________________________________________________ Matt University crawford@anl-mcs.arpa Crawford of Chicago ihnp4!oddjob!matt
cjh@petsd.UUCP (Chris Henrich) (08/27/85)
[] In article <946@oddjob.UUCP> matt@oddjob.UUCP (Matt Crawford) writes: >... the number 221*5^n seems to be expressible as a sum >of two squares 2n+2 different ways for n through at least 10 or >so. Is it true for all n? If so, why? This kind of question has been extensively studied by number theorists. References: many many textbooks on number theory. That by Hardy (and Littlewood?) is a standard reference. There is also Cohn, _A_Second_Course_in_Number_Theory_ and Serre _Arithmetic_ (don't be fooled, this is *advanced*). Here is an ad-hoc proof of yor observation, based on Gaussian integers (i.e. complex numbers, whose real and imaginary parts are integers). _ 221 = Z * Z where Z = (14 + 5 i) _ 5 = W * W where W = ( 2 + i ) therefore _ 221 * 5^n = U * U where _ U = Z * W^a * W^b a + b = n. Now the real and imaginary parts of any such U, in either order, are a solution to your equation. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288
matt@oddjob.UUCP (Matt Crawford) (08/27/85)
Thanks for the proof outline, Christopher. I see that the
hardest part is left out, though -- counting the duplicate
solutions for U's that are conjugate or related by factors
of i. Is the condition something along the line of all
_
the Z's and Z's not being divisible (over the gaussian
_
integers) by W or W?
_____________________________________________________
Matt University crawford@anl-mcs.arpa
Crawford of Chicago ihnp4!oddjob!mattjankok@zuring.UUCP (08/28/85)
In article 170 Bill Sinclair reminds us of the story of Ramanujan,
discussing his room number 1729 with a friend.
The story is told in the following way by Newman (in J.R. Newman (ed.)
The world of Mathematics, Vol. I, p. 375, Simon and Schuster, 1956).
Newman quoted Hardy, probably from his obituary of Ramanujan, who wrote
that he rode in a taxi with number 1729 to Ramanujan who lay ill at
Putney. When arrived they discussed the number.
I do not know a solution to the question. I expect that numbers exist
which can be split three times, if the amount of numbers which can
be split twice is large or infinite.pumphrey@ttidcb.UUCP (Larry Pumphrey) (08/28/85)
> Misspending part of a summer on this sort of question led me to > observe that the number 221*5^n seems to be expressible as a sum > of two squares 2n+2 different ways for n through at least 10 or > so. Is it true for all n? If so, why? Yes, it is true for _all_ n. In fact the more general problem can be phrased as follows: n _________ |\ | | | |/ | \ | | | _ |\i Let | \ | = | | |_) | \ | | | | i | \| | | i=1 where each p is a prime of the form 4x+1 and further i let f(N) be the number of _different_ divisors of N (1 is considered a divisor) less than or equal to the square root of N. Then N can be represented as the sum of 2 squares in exactly f(N) ways. ----------------------------------------------------------------------- First proof wins a box of cheerios. My proof is over 200 lines long so I'm not posting at this time --- margin is too small to contain it :-) Hint: It can be solved by elementary means, that is to say algebraically rather than analytically! p.s. Don't worry about misspending a summer on this problem, I've wasted my whole life (48 and still counting) trying to prove x^n + y^n = z^n has no integral solutions for n>2 :-( Enjoy!
cjh@petsd.UUCP (Chris Henrich) (08/29/85)
[] In article <949@oddjob.UUCP> matt@oddjob.UUCP (Matt Crawford) writes: >Thanks for the proof outline, Christopher. I see that the >hardest part is left out, though -- counting the duplicate >solutions for U's that are conjugate or related by factors >of i. Is the condition something along the line of all > _ >the Z's and Z's not being divisible (over the gaussian > _ >integers) by W or W? Yes. And that there is "unique factorization" in the Gaussian integers. By the way, the attempt to generalize these ideas to other quadratic forms was carried out by Gauss, and led to the modern theory of ideals in rings of algebraic integers. Part of that theory, called "class field theory", is very tough stuff. Regards, Chris -- Full-Name: Christopher J. Henrich UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288
msb@lsuc.UUCP (Mark Brader) (08/29/85)
> Most of you have probably heard the story of Ramanujan, who was riding > in the cab with a friend. They were discussing his room number 1729, > when his friend remarked that it was an uninteresting number. > "Oh no" Ramanujan replied. "it is the smallest number that can be written > as the sum of two cubes in two different ways". Actually, Hardy took CAB number 1729 to visit Ramanujan. The interesting thing here is that 1729 is even more interesting than Ramanujan mentioned. It is not the smallest number to have a certain other property, but it IS the THIRD-smallest, and that makes the property pretty rare. Combining this property with the UNRELATED one that Ramanujan mentioned makes 1729 very interesting indeed! What property am I talking about? I'll post the answer in a few days if I don't see it posted first. Mark Brader
paulv@boring.UUCP (09/01/85)
In article <388@aero.ARPA> sinclair@aero.UUCP (William S. Sinclair) writes: > >Most of you have probably heard the story of Ramanujan, who was riding >in the cab with a friend. They were discussing his room number 1729, >when his friend remarked that it was an uninteresting number. >"Oh no" Ramanujan replied. "it is the smallest number that can be written >as the sum of two cubes in two different ways". > > Bill Sinclair (asbestos Willie) According to C.P. Snow in the preface to G.H. Hardy's `A Mathematicians Apology': Hardy used to visit him as he lay dying in hospital at Putney. [He] always inept about introducing a conversation, said, probably without greeting, and certainly as his first remark,: `I thought the number of my taxicab was 1729. It seemed to me a rather dull number.' To which Ramanujan replied: `No Hardy! no Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.'
rpb@aaec.OZ (Bob Backstrom) (09/02/85)
> > Most of you have probably heard the story of Ramanujan, who was riding > in the cab with a friend. They were discussing his room number 1729, > when his friend remarked that it was an uninteresting number. > "Oh no" Ramanujan replied. "it is the smallest number that can be written > as the sum of two cubes in two different ways". > > My question is, what is the smallest number that can be written as the > sum of two cubes in THREE different ways? Does one exist? > > Bill Sinclair (asbestos Willie) [ I am posting this reply to the net after getting Unknown Host when mailing to "sinclair@aero.ARPA" ] I had looked at exactly this problem some years ago and found the first four such solutions as follows: 3 3 87,539,319 = 167 + 436 3 3 = 228 + 423 3 3 = 255 + 414 3 3 119,824,488 = 11 + 493 3 3 = 90 + 492 3 3 = 346 + 428 3 3 143,604,279 = 111 + 522 3 3 = 359 + 460 3 3 = 408 + 423 3 3 175,959,000 = 70 + 560 3 3 = 198 + 552 3 3 = 315 + 525 From Bob Backstrom at the Australian Atomic Energy Commission, Sydney, New South Wales, Australia.
rhm@bonnie.UUCP (Bob Morris) (09/03/85)
> > > Misspending part of a summer on this sort of question led me to > > observe that the number 221*5^n seems to be expressible as a sum > > of two squares 2n+2 different ways for n through at least 10 or > > so. Is it true for all n? If so, why? > > Yes, it is true for _all_ n. In fact the more general problem can be > phrased as follows: > > n > _________ > |\ | | | |/ > | \ | | | _ |\i > Let | \ | = | | |_) > | \ | | | | i > | \| | | > i=1 > > > where each p is a prime of the form 4x+1 and further > i > let f(N) be the number of _different_ divisors of N (1 is > considered a divisor) less than or equal to the square root of N. > > Then N can be represented as the sum of 2 squares in exactly f(N) ways. > > ----------------------------------------------------------------------- > > First proof wins a box of cheerios. My proof is over 200 lines long so > I'm not posting at this time --- margin is too small to contain it :-) > Hint: It can be solved by elementary means, that is to say algebraically > rather than analytically! > > p.s. Don't worry about misspending a summer on this problem, I've > wasted my whole life (48 and still counting) trying to prove > > x^n + y^n = z^n > > has no integral solutions for n>2 :-( > Enjoy! *** REPLACE THIS LINE WITH YOUR MESSAGE *** Spending a summer to come up with 221*5^n is overkill. It works as stated and the proof is available in any non-trivial book on number theory. But 13*5^n works equally well, and, indeed, 5^n is not bad either. On another subject that came up recently, there are numbers that can be expressec in k ways as the sum of two cubes, for any value of k. The proof is (strangely enougn) constructed and can be found somewhere toward the end of Hardy and Wright.
msb@lsuc.UUCP (Mark Brader) (09/03/85)
> The interesting thing here is that 1729 is even more interesting than > Ramanujan mentioned. It is not the smallest number to have a certain > other property, but it IS the THIRD-smallest, and that makes the property > pretty rare. Combining this property with the UNRELATED one that > Ramanujan mentioned makes 1729 very interesting indeed! > What property am I talking about? The answer is: 1729 is also a Carmichael number. Fermat proved that b^p - b is divisible by p for all positive(?) integers b, if p is prime. But it's "if", not "if any only if". A composite number p having the same property is called a Carmichael number. According to a program I ran, the ones below 65536 are: 561 = 3x11x17 1105 = 5x13x17 1729 = 7x13x19 2465 = 5x17x29 2821 = 7x13x31 6601 = 7x23x41 8911 = 7x19x67 10585 = 5x29x73 15841 = 7x31x73 29341 = 13x37x61 41041 = 7x11x13x41 My reference on this is the December 1982* Scientific American, the article on "The Search for Prime Numbers". (The article said that 561 was the first of these numbers and then casually mentioned 1729 as another. I liked that.) Mark Brader *Oops, I should have written down the year. It might be 1983.