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A338445
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Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with 1<=x<m.
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2
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3, 11, 13, 19, 23, 29, 31, 43, 49, 53, 57, 59, 61, 67, 71, 73, 77, 79, 83, 85, 89, 91, 93, 97, 101, 103, 109, 113, 127, 129, 131, 133, 141, 143, 147, 149, 151, 157, 161, 163, 167, 169, 173, 177, 179, 183, 187, 197, 199, 201, 203, 205, 211, 217, 229, 235, 237, 239
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OFFSET
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1,1
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COMMENTS
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Some values of m have multiple solutions.
For example, for m = 49, 25^25 == 26^26 (mod 49) and 37^37 == 38^38 (mod 49).
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LINKS
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EXAMPLE
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3 is a term because 1^1 == 2^2 (mod 3).
11 is a term because 8^8 == 9^9 (mod 11).
13 is a term because 8^8 == 9^9 (mod 13).
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MAPLE
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filter:= proc(n) local x, y, z;
y:= 1;
for x from 2 to n-1 do
z:= x &^ x mod n;
if z = y then return true fi;
y:= z
od;
false
end proc:
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MATHEMATICA
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seqQ[n_] := AnyTrue[Range[n - 1], PowerMod[#, #, n] == PowerMod[# + 1, # + 1, n] &]; Select[Range[240], seqQ] (* Amiram Eldar, Oct 28 2020 *)
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PROG
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(PARI) isok(m)=sum(i=1, m-1, Mod(i, m)^i == Mod((i+1), m)^(i+1)) \\ Andrew Howroyd, Oct 28 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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