Diophantine Analysis
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It is obvious that one can obtain immediately an w-param- eter solution of the equation Pix)=l\ where k is any positive integer. For this purpose it is suffi- cient to write l = P{z) and to determine xi, . . ., Xn from the relation {P{z)\' = Pix).
+ oci V„, 0, by its V alue 52 DIOPHANTINE ANALYSIS Again, from a single solution of the equation . P(x)=o, an w-parameter solution may be obtained by means of Eq.
(ii), yi, . . ., yn being taken as arbitrary and the quantities Zi, . . ., Zn being the ...new values of xi, . . ., Xn.
In a similar manner, from a single solution of the equation P(x) = i. (12) others may be found. From a solution of Eq. (12) and a solution of the equation P{x)^7n, other solutions of the latter may obviously be obtained.
GENERAL EXERCISES 1. li p, q, r satisfy the relations 1,1,1 p+q+r = i, -+-+-=0, p q r show that a^+b^+c^^(pa+qb+rc)-+{qa+rb+pc)-+ira+pb+qcy.
(Davis, 191 2.) 2. Obtain solutions of the equation {x-+y~+z'^){xi^+yi'^+zr) = ti^+v'^+iv'^.
(Catalan, 1893.) 3.
What to read after Diophantine Analysis?
You can find similar books in the "Read Also" column, or choose other free books by R D Robert Daniel Carmichael to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
It is obvious that one can obtain immediately an w-param- eter solution of the equation Pix)=l\ where k is any positive integer. For this purpose it is suffi- cient to write l = P{z) and to determine xi, . . ., Xn from the relation {P{z)\' = Pix).
+ oci V„, 0, by its V alue 52 DIOPHANTINE ANALYSIS Again, from a single solution of the equation . P(x)=o, an w-parameter solution may be obtained by means of Eq.
(ii), yi, . . ., yn being taken as arbitrary and the quantities Zi, . . ., Zn being the ...new values of xi, . . ., Xn.
In a similar manner, from a single solution of the equation P(x) = i. (12) others may be found. From a solution of Eq. (12) and a solution of the equation P{x)^7n, other solutions of the latter may obviously be obtained.
GENERAL EXERCISES 1. li p, q, r satisfy the relations 1,1,1 p+q+r = i, -+-+-=0, p q r show that a^+b^+c^^(pa+qb+rc)-+{qa+rb+pc)-+ira+pb+qcy.
(Davis, 191 2.) 2. Obtain solutions of the equation {x-+y~+z'^){xi^+yi'^+zr) = ti^+v'^+iv'^.
(Catalan, 1893.) 3.
What to read after Diophantine Analysis?
You can find similar books in the "Read Also" column, or choose other free books by R D Robert Daniel Carmichael to read onlineMoreLess
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