There are infinite (real) solutions for: a^3 + b^3 + (43 - a - b)^3 = 17299^2
Basically you can replace *b* with any number and you have a quadratic equation (because a^3 will go away)... Any quadratic equation has 2 solutions (might give the same value, and both are complex or real). -3ba^2 + 129a^2 - 3ab^2 + 258ab - 5547a + 129b^2 - 5547b - 299175894 = 0 If *b* was a constant then we have: a^2(-3b+129) + a (-3b^2+258b-5547) + (129b^2 - 5547b - 299175894) = 0 (-3b^2+258b-5547)^2 - 4 * (-3b+129) * (129b^2 - 5547b - 299175894) >= 0 is what we need to have both real solutions (if I didn't make any silly mistake) and that has infinite solutions (*b* <= 43 or *b* >= ~722.943). Best, Diego On Sat, Jul 16, 2011 at 15:32, Alfonso J. Ramos <[email protected]> wrote: > Excuse me (I used spanish): not in the integers, but apparently there is > one in the reals: > > http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a^3+%2B+b^3+%2B+c^3+%3D+17299^2<http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a%5E3+%2B+b%5E3+%2B+c%5E3+%3D+17299%5E2> > > 2011/7/16 Alfonso J. Ramos <[email protected]> > > No en los enteros, pero al parecer si en los reales: >> >> http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a^3+%2B+b^3+%2B+c^3+%3D+17299^2<http://www.wolframalpha.com/input/?i=+a+%2B+b+%2B+c+%3D+43+and+a%5E3+%2B+b%5E3+%2B+c%5E3+%3D+17299%5E2> >> >> >> 2011/7/16 Luke Pebody <[email protected]> >> >>> No answers exist for a + b + c = 43 and a^3 + b^3 + c^3 = 17299^2, >>> even allowing them all to be negative: >>> http://ideone.com/FTm8l >>> >>> On Sat, Jul 16, 2011 at 6:28 PM, Leopoldo Taravilse >>> <[email protected]> wrote: >>> > You can solve it with no programming. >>> > a+b+c = 43 && a^3 + b^3 + c^3 = 17299 >>> > As 17299 == 3 mod 4 then as a^3,b^3 and c^3 can be 0 or 1 mod 4 then a, >>> b >>> > and c are odd. >>> > 27^3 > 17299 so a < b < c <= 25 >>> > 1^3 == 1 mod 5 >>> > 2^3 == 3 mod 5 >>> > 3^3 == 2 mod 5 >>> > 4^3 == 4 mod 5 >>> > 5^3 == 5 mod 5 (or 0^3 == 0 mod 5) >>> > So as 17299 == 43 + 1 mod 5 at least one of a, b and c must be 2 mod 5 >>> so >>> > there are two options, 7 or 17. >>> > Case 1 (7): a+b = 36 && a^3 + b^3 = 16956 >>> > Case 2 (17) a+b = 26 && a^3 + b^3 = 12386 >>> > Let's see Case 1 first: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, >>> 21, >>> > 23 or 25, but if a = 1, 3, 5 or 9 then b > 25 so a and b can be 11, 13, >>> 15, >>> > 17, 19, 21, 23 or 25. We can easily discard 13 and 23 because of modulo >>> 5 >>> > reasons so we have 11, 15, 17, 19, 21 or 25 so we can try with this six >>> > numbers. >>> > Case 2: a and b can be 1, 3, 5, 9, 11, 13, 15, 17, 19, 21 or 23, but we >>> can >>> > discard 3, 13 and 23 again so we have 1, 5, 9, 11, 15, 17, 19 and 21 to >>> try. >>> > >>> > On Sat, Jul 16, 2011 at 6:54 AM, micke <[email protected]> wrote: >>> >> >>> >> What are the three positive integers whose sum is 43. and the sum of >>> >> the cubes of three integers is square of the number 17299. >>> >> >>> >> -- >>> >> You received this message because you are subscribed to the Google >>> Groups >>> >> "google-codejam" group. >>> >> To post to this group, send email to [email protected]. >>> >> To unsubscribe from this group, send email to >>> >> [email protected]. >>> >> For more options, visit this group at >>> >> http://groups.google.com/group/google-code?hl=en. >>> >> >>> > >>> > -- >>> > You received this message because you are subscribed to the Google >>> Groups >>> > "google-codejam" group. >>> > To post to this group, send email to [email protected]. >>> > To unsubscribe from this group, send email to >>> > [email protected]. >>> > For more options, visit this group at >>> > http://groups.google.com/group/google-code?hl=en. >>> > >>> >>> -- >>> You received this message because you are subscribed to the Google Groups >>> "google-codejam" group. >>> To post to this group, send email to [email protected]. >>> To unsubscribe from this group, send email to >>> [email protected]. >>> For more options, visit this group at >>> http://groups.google.com/group/google-code?hl=en. >>> >>> >> > -- > You received this message because you are subscribed to the Google Groups > "google-codejam" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/google-code?hl=en. > -- You received this message because you are subscribed to the Google Groups "google-codejam" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/google-code?hl=en.
