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Smallest even pseudoprime (>2n+1) in base 2n+1.
1

%I #39 Feb 16 2025 08:33:24

%S 4,286,124,16806,28,70,244,742,1228,906,1852,154,28,286,52,66,496,442,

%T 66,1834,344,526974,76,506,66,70,286,1266,2296,946,130,5662,112,154,

%U 14246,370,276,8614,2806,2626,112,1558,276,2626,19126,1446,322,658,176,742,190,946,5356,742,186,190,176,8474,2806,2242,148

%N Smallest even pseudoprime (>2n+1) in base 2n+1.

%C For an even base there are no even pseudoprimes.

%C Conjecture: There are infinitely many even pseudoprimes in every odd base.

%C Records: 4, 286, 16806, 526974, 815866, 838246, ..., and they occur at indices: 0, 1, 3, 21, 503, 691, ...

%H Eric Chen, <a href="/A253233/b253233.txt">Table of n, a(n) for n = 0..999</a> (a(0) corrected by _Georg Fischer_, Jan 20 2019)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPseudoprime.html">Fermat pseudoprime</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermat_pseudoprime">Fermat pseudoprime</a>

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%F a(A005097(n-1)) = A108162(n).

%t f[n_] := Block[{k = 2 * n + 2}, While[PrimeQ[k] || OddQ[k] || PowerMod[2 * n + 1, k - 1, k] != 1, k++ ]; k]; Table[ f[n], {n, 0, 60}]

%o (PARI) a(n) = for(k=n+1, 2^24, if(!isprime(2*k) && Mod(2*n+1, 2*k)^(2*k-1) == Mod(1, 2*k), return(2*k)))

%Y Cf. A005097, A090082, A090083, A090084, A090085, A090086, A090087, A090088, A090089, A108162, A130433, A130434, A130435, A130436, A130437, A130438, A130439, A130440, A139441, A130442, A130443, A007535.

%K nonn,changed

%O 0,1

%A _Eric Chen_, May 17 2015