I am a mathematical hobbyist puttering around with MVR4 (I know, dated software....) exploring some elementary number theory. In particular, I am trying to get a better grasp of the classic Dirichlet proofs for FLT with n = 5, which is basically high school algebra very ingeniously applied.

I am looking at numbers of the form \(a + b \sqrt {5}\). On raising this to the ﬁfth power and expanding things out I get something that is correct but not compact.

The result is an expression with three terms in a and b and three terms in a, b, and sqrt(5).

How do I tell Maple to collect the terms so that I get the result in this form:

(sum of terms without ` sqrt(5)) + (sum of "coefficients" of terms with sqrt(5)) * sqrt(5)`

,

i.e. a number of the form `A + B*sqrt(5)`

, where A and B are sums of terms in a and b
?

Well, one thing you can do is substitute a symbol for sqrt(5), collect with respect to that, and substitute back. For example:

> R:= (your expression); > subs(sq5=sqrt(5),collect(subs(sqrt(5)=sq5,R),sq5));

This is not at all a trivial task. Below you will ﬁnd two possible solutions for MapleV/Release4.
Each of them works with algebraic numbers of type `a+b*sqrt(D)`

or `a+b*sqrt(-D)`

. The ﬁrst
approach does not work within procedures.

Solution 1

> restart; > alias(alpha=sqrt(5)); > x1:=a+b*alpha; > x1e:=sort(frontend(collect,[expand(x1^5),alpha])); 5 3 2 4 4 2 3 5 x1e := a + 50 a b + 125 a b + (5 a b + 50 a b + 25 b ) alpha > x2:=a+b*I*alpha; x2 := a + I b alpha > x2e:=sort(frontend(collect,[expand(x2^5),alpha])); 5 3 2 4 x2e := a - 50 a b + 125 a b 4 2 3 5 + (5 I a b - 50 I a b + 25 I b ) alpha > alias(alpha=alpha); > rules:={I='i',sqrt(5)=sqrt5}; > invrules:=map(u->op(2,u)=op(1,u),rules); > subs(invrules,applyop(factor,-1,subs(rules,x1e))); 5 3 2 4 4 2 2 4 1/2 a + 50 a b + 125 a b + 5 b (a + 10 a b + 5 b ) 5 > subs(invrules,applyop(factor,-1,subs(rules,x2e))); 5 3 2 4 4 2 2 4 1/2 a - 50 a b + 125 a b + 5 I b (a - 10 a b + 5 b ) 5

Solution 2

> restart; > collecta:=proc(e,K) if has(K,I) then evalc(expand(e)); %-op(-1,%)+map(u->frontend(factor,[u]),op(-1,%)/(K/I))*(K/I) else sort(frontend(collect,[expand(e),K])); map(u->frontend(factor,[u]),%) fi end: > x1:=a+b*sqrt(5); > collecta(x1^5,sqrt(5)); 5 3 2 4 4 2 2 4 1/2 a + 50 a b + 125 a b + 5 b (a + 10 a b + 5 b ) 5 > x2:=a+b*sqrt(-5); > collecta(x2^5,sqrt(-5)); 5 3 2 4 4 2 2 4 1/2 a - 50 a b + 125 a b + 5 I b (a - 10 a b + 5 b ) 5

One approach is to use "remove" for the ﬁrst sum and "select" for the second sum (note the trick to extract sqrt(5)):

> restart: > w:=sqrt(5); > f:=a+b*w; > e:=expand(f^5); 5 4 1/2 3 2 2 3 1/2 4 e := a + 5 a b 5 + 50 a b + 50 a b 5 + 125 a b 5 1/2 + 25 b 5 > remove(has,e,w)+w*expand(select(has,e,w)/w); 5 3 2 4 1/2 4 2 3 5 a + 50 a b + 125 a b + 5 (5 a b + 50 a b + 25 b )