Prime Sierpiński Numbers of the First Kind
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Open Question
The only known prime Sierpiński numbers of the first kind are:
\(\ds 1^1 + 1\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds 2^2 + 1\) | \(=\) | \(\ds 5\) | ||||||||||||
\(\ds 4^4 + 1\) | \(=\) | \(\ds 257\) |
This sequence is A121270 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
It is an open question as to whether there are any more.
Progress
Let $S_n$ be the $n$th Sierpiński number of the first kind:
- $S_n = n^n + 1$
From Form of Prime Sierpiński Number of the First Kind, it is known that if $S_n$ is prime, then:
- $n = 2^{2^k}$
for some integer $k$.
Thus
\(\ds S_n\) | \(=\) | \(\ds \left({2^{2^k} }\right)^{\left({2^{2^k} }\right)} + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{2^k \times \left({2^{2^k} }\right)} + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{2^{k + 2^k} } + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{2^m} + 1\) | where $m = k + 2^k$ |
Thus a prime $S_n$ is a Fermat number $F_m$ where $m = k + 2^k$.
The sequence of $m$ begins:
- $1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, \ldots$
This sequence is A006127 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Thus it remains to investigate the following Sierpiński numbers of the first kind for primality:
$k$ $m = k + 2^k$ $n = 2^{2^k}$ Status of $S_n = n^n + 1 = 2^{2^m} + 1 = F_m$ $0$ $1$ $2$ Prime ($S_2 = F_1 = 5$) $1$ $3$ $4$ Prime ($S_4 = F_3 = 257$) $2$ $6$ $16$ Composite with factor $1071 \times 2^8 + 1$ $3$ $11$ $256$ Composite with factor $39 \times 2^{13} + 1$ $4$ $20$ $65 \, 536$ Composite with no factor known $5$ $37$ $4 \, 294 \, 967 \, 296$ Composite with factor $1 \, 275 \, 438 \, 465 \times 2^{39} + 1$ $6$ $70$ too large unknown $7$ $135$ unknown $8$ $264$ unknown $9$ $521$ unknown $10$ $1034$ unknown $11$ $2059$ Composite with factor $591 \, 909 \times 2^{2063} + 1$ $12$ $4108$ unknown $13$ $8205$ unknown $14$ $16 \, 398$ unknown $15$ $32 \, 783$ unknown $16$ $65 \, 552$ unknown $17$ $131 \, 089$ unknown
Sources
- 1966: Joseph S. Madachy: Mathematics on Vacation
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $257$
- Weisstein, Eric W. "Sierpiński Number of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SierpinskiNumberoftheFirstKind.html