Question 1204795

{{{b=(a-1)^1/2 +1 }}}
 {{{c=(b-1)^1/2 +1 }}}
 {{{d=(c-1)^1/2 +1 }}}
{{{e= (d-1)^1/2 +1 }}}


Find the least possible value of {{{a +b +c +d +e}}}


A Fermat number is of the form



*[tex \Large F_n\ =\ 2^2^n\,+\,1].
 

{{{a}}}, {{{b}}}, {{{c}}}, {{{d}}}, {{{e}}} are {{{F[n]}}} for {{{n}}} = { {{{1}}},  {{{2}}},  {{{3}}},  {{{4}}} }

*[tex \Large F_n\ =\ 2^2^4\,+\,1=65537]

*[tex \Large F_n\ =\2^2^3\,+\,1=257]

*[tex \Large F_n\ =\2^2^2\,+\,1=17]

*[tex \Large F_n\ =\2^2^1\,+\,1=5]

*[tex \Large F_n\ =\2^2^0\,+\,1=3]


so, {{{a}}}, {{{b}}}, {{{c}}}, {{{d}}}, and {{{e}}} are the five known prime Fermat numbers (in descending order)

{{{65537}}}, {{{257}}},{{{ 17}}}, {{{5}}}, and {{{3}}}.


their sum is:

{{{65537+257+17+5+3=65819}}}


answer:

D) {{{65819}}}