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10-adic integer x=...07839804103263499879186432 satisfying x^5 = x; also x^3 = -x = A120818; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.
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%I #35 Aug 31 2021 04:35:22

%S 2,3,4,6,8,1,9,7,8,9,9,4,3,6,2,3,0,1,4,0,8,9,3,8,7,0,4,0,3,5,5,6,1,4,

%T 2,2,1,4,4,1,5,4,2,3,0,3,5,5,4,0,3,3,2,2,3,2,5,9,4,6,9,3,8,3,9,5,2,6,

%U 8,6,0,9,5,7,2,0,9,1,4,6,4,3,6,4,9,6,3,3,3,0,8,2,0,3,3,5,8,8,3,4,0,4,3,5,5

%N 10-adic integer x=...07839804103263499879186432 satisfying x^5 = x; also x^3 = -x = A120818; (x^2)^3 = x^2 = A091664; (x^4)^2 = x^4 = A018248.

%H Seiichi Manyama, <a href="/A120817/b120817.txt">Table of n, a(n) for n = 0..9999</a> (terms 0..999 from Paul D. Hanna)

%H Patrick A. Thomas, <a href="/A120817/a120817_1.txt">Solutions to x^5=x up to base 100</a>

%F x = 10-adic limit_{n->infinity} 2^(5^n).

%e x equals the limit of the (n+1) trailing digits of 2^(5^n):

%e 2^(5^0)=(2), 2^(5^1)=(32), 2^(5^2)=33554(432),

%e 2^(5^3)=4253529586511730793292182592897102(6432), ...

%e x=...93839649523223304553032451441224165530407839804103263499879186432.

%e x^2=...0557423423230896109004106619977392256259918212890624 (A091664).

%e x^3=...6695446967548558775834469592160195896736500120813568 (A120818).

%e x^4=...9442576576769103890995893380022607743740081787109376 (A018248).

%e x^5=...3304553032451441224165530407839804103263499879186432 = x.

%o (PARI) {a(n)=local(b=2,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]}

%o (PARI) {a(n)=if(n<0, 0, lift(chinese(Mod(truncate( teichmuller(2+O(5^(n+1)))), 5^(n+1)), Mod(0, 2^(n+1))))\10^n)} /* _Michael Somos_, Oct 03 2006 */

%Y x^5 = x: this sequence (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), A290374 (...7057), A290375 (...4193).

%Y Cf. A091664, A018248.

%K base,nonn

%O 0,1

%A _Paul D. Hanna_, Jul 06 2006