Answer to Question #146606 in Combinatorics | Number Theory for ankit

Question #146606
Let us denote S_n=a^n+b^n+c^n for arbitrary numbers a,b,c. It is known that S_1=4,5, S_2=22,25, S_3=104,625 for some values of a,b,c. What is the largest possible value of S_{737}^2-S_{736}S_{738}?
1
Expert's answer
2020-12-03T07:51:06-0500

S7372-S736*S738=(a737+b737+c737)2-(a736+b736+c736)(a738+b738+c738)=

a1474+a737b737+a737c737+b737a737+b1474+b737c737+c737a737+c737b737+c1474-

-(a1474+a736b738+a736c738+b736a738+b1474+b736c738+c736a738+c736b738+c1474)=

2a737b737+2a737c737+2b737c737-a736(b738+c738)-b736(a738+c738)-c736(a738+b738)

It is known that:

4.5=a+b+c

22.25=a2+b2+c2

104.625=a3+b3+c3

Since the system is symmetric we can consider only one solution:

c=4.5-a-b

22.25=a2+b2+(4.5-a-b)2

b="\\frac{1}{4}(-\\sqrt{-12*a^2+36*a+97}-2a+9)"

104.625=a3+"(\\frac{1}{4}(-\\sqrt{-12*a^2+36*a+97}-2a+9))^3"+"(4.5-a-\\frac{1}{4}(-\\sqrt{-12*a^2+36*a+97}-2a+9))^3"

a=0

b="\\frac{9}{4}-\\frac{\\sqrt{97}}{4}"

c="\\frac{9}{4}+\\frac{\\sqrt{97}}{4}"


S7372-S736*S738=2b737c737-b736(c738)-c736(b738)=2"(\\frac{9}{4}-\\frac{\\sqrt{97}}{4})^{737}" * "(\\frac{9}{4}+\\frac{\\sqrt{97}}{4})^{737}" -"(\\frac{9}{4}-\\frac{\\sqrt{97}}{4})^{736}" "(\\frac{9}{4}+\\frac{\\sqrt{97}}{4})^{738}" -"(\\frac{9}{4}-\\frac{\\sqrt{97}}{4})^{738}" "(\\frac{9}{4}+\\frac{\\sqrt{97}}{4})^{736}" =-2-"(\\frac{9}{4}+\\frac{\\sqrt{97}}{4})^{2}"-"(\\frac{9}{4}-\\frac{\\sqrt{97}}{4})^{2}"=

-2-"\\frac{1}{8}(89+9\\sqrt{97})" -"\\frac{1}{8}(89-9\\sqrt{97})" =-2-89/8-89/8=2-89/4=-"\\frac{81}{4}"


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