Intrigued by an challenge in the Dutch Math Olympiade, I studied a way to evaluate infinite nested radicals.
The question was to evaluate $\sqrt{4+\sqrt{18+\sqrt{40+\sqrt{70+\sqrt{108+\sqrt{156+…...}}}}}}$
It is easily converted to the general form $${I_n}^2=an^2+bn+c+(d+en) \cdot I_{n+1}$$ There can be only a solution ${I_n}=p+qn$ when all powers match. This restricts the possibilities. By choosing p and q (which determines or at least restricts a,b,c,d and e) I was able to derive a large collection of infinite nested radicals with an integer as result. Sometimes truncations or extension of another infinite nested radical are found. I have listed below only the ‘primitive’ forms where all terms are positive and the first term in the root is as small as possible. In this way I derived: $$2=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+…...}}}}}}$$ $$2=\sqrt{1+\sqrt{5+\sqrt{11+\sqrt{19+\sqrt{29+\sqrt{41+…...}}}}}}$$ $$3=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+…...}}}}}}$$ $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+…...}}}}}}$$ $$3=\sqrt{1+\sqrt{51+\sqrt{151+\sqrt{301+\sqrt{501+\sqrt{751+…...}}}}}}$$ $$3=\sqrt{2+\sqrt{38+\sqrt{106+\sqrt{206+\sqrt{338+\sqrt{502+…...}}}}}}$$ $$3=\sqrt{3+\sqrt{9+\sqrt{27+\sqrt{69+\sqrt{129+\sqrt{207+…...}}}}}}$$ $$3=\sqrt{4+1\sqrt{4+3\sqrt{4+5\sqrt{4+7\sqrt{4+9\sqrt{4+…...}}}}}}$$ $$3=\sqrt{4+\sqrt{18+\sqrt{40+\sqrt{70+\sqrt{108+\sqrt{156+…...}}}}}}$$ $$4=\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12+…...}}}}}}$$ $$4=\sqrt{4+2\sqrt{4+4\sqrt{4+6\sqrt{4+8\sqrt{4+10\sqrt{4+…...}}}}}}$$ $$4=\sqrt{7+\sqrt{67+\sqrt{177+\sqrt{377+\sqrt{547+\sqrt{807+…...}}}}}}$$ $$4=\sqrt{8+\sqrt{52+\sqrt{128+\sqrt{236+\sqrt{376+\sqrt{548+…...}}}}}}$$ $$4=\sqrt{9+\sqrt{39+\sqrt{87+\sqrt{153+\sqrt{237+\sqrt{339+…...}}}}}}$$ $$4=\sqrt{10+\sqrt{28+\sqrt{54+\sqrt{88+\sqrt{130+\sqrt{180+…...}}}}}}$$ $$5=\sqrt{15+\sqrt{85+\sqrt{205+\sqrt{375+\sqrt{595+\sqrt{865+…...}}}}}}$$ $$5=\sqrt{16+\sqrt{68+\sqrt{152+\sqrt{268+\sqrt{416+\sqrt{596+…...}}}}}}$$ $$5=\sqrt{17+\sqrt{53+\sqrt{107+\sqrt{153+\sqrt{179+\sqrt{269+…...}}}}}}$$ $$6=\sqrt{25+\sqrt{105+\sqrt{235+\sqrt{415+\sqrt{645+\sqrt{925+…...}}}}}}$$ $$6=\sqrt{26+\sqrt{86+\sqrt{178+\sqrt{302+\sqrt{458+\sqrt{646+…...}}}}}}$$ $$7=\sqrt{37+\sqrt{127+\sqrt{267+\sqrt{457+\sqrt{697+\sqrt{987+…...}}}}}}$$
I wonder if these results are known. (Some of them are published before, like the one, once asked and answered by Ramanujan)
