By James Matteson
1 Diophantine challenge, it truly is required to discover 4 affirmative integer numbers, such that the sum of each of them can be a dice. resolution. If we think the first^Cx3^)/3-), the second^^x3-y3--z* ), the third=4(-z3+y3+*'), and the fourth=ws-iOM"^-*)5 then> the 1st further to the second=B8, the 1st extra to the third=)/3, the second one additional to third=23, and the 1st extra to the fourth=ir therefore 4 of the six required stipulations are chuffed within the notation. It is still, then, to make the second one plus the fourth= v3-y3Jrz*=cnbe, say=ic3, and the 3rd plus the fourth^*3- 23=cube, say=?«3. Transposing, we need to unravel the equalities v3--£=w3--if=u?--oi?; and with values of x, y, z, in such ratio, that every will probably be more than the 3rd. allow us to first unravel, mostly phrases, the equality «'-}-23=w3-|-y3. Taking v=a--b, z=a-b, w-c--d, y=c-d, the equation, after-dividing via 2, turns into a(a2-)-3i2)==e(c2-J-3f72). Now suppose a-Sn])--Smq, b=mp-3nq, c=3nr
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Extra resources for A collection of Diophantine problems with solutions
Now any polynomial p(s) can be expanded as a finite product p(s) =p(0) [1 - (s/p)],where p ranges over the roots of the equation p ( p ) = 0 [except that the product formula for p(s) is slightly different if p(0) = 01; hence the product formula (4) states that &(s) is like a polynomial of inznite degree. (Similarly, Euler thought of sin x as a “polynomial of infinite degree” when he conjectured, and finally ] ] . the ) other hand, the proved, the formula sinnx = nx JJ;p=l [l - ( ~ / n ) ~On statement that the series (3) converges “very rapidly” is also a statement that t(s) is like a polynomial of infinite degree-a finite number of terms gives a very good approximation in any finite part of the plane.
To evaluate the limit of G(B), set ~3= n iz, where c is fixed and z + 00. The’change of variable t = eu, u = log t puts G(B) in the form - + rx ib + loax e,9u ib-m -du+ u ib+loax eS‘ -du, U where the path of integration has been altered slightly using Cauchy’s theorem. The changes of variable u = is v in the first integral and u = log x iw in the second put this in the form + + 6 G(B) = d6ue-d7 ea i w dw . logx iw e-7w + Both integrals in this expression approach zero as z -+ 00, the first because e-d7-+ 0 and the second because e-rw-+ 0 except at w = 0.
Therefore f(s) = 0 which is constant, as was to be shown. 8 THE PRODUCT FORMULA FOR 4 The function F(s) = &)/IT, [l - (s - i ) / ( p - i)] is analytic in the entire s-plane and is an even function of s - 4. Moreover, it has no zeros, so its logarithm is well defined up to an additive constant 2nni (n an integer) by the formula log F(s) = Ji F'(z) dz/F(z) log F(O), where log F(0) is determined to within an additive constant 2nni. The results of the preceding two sections then combine to give log F(s) = const, and therefore upon exponentiation + where c is a constant.