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A278573
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Irregular triangle read by rows: row n lists values of k in range 1 <= k <= n-1 such x^n + x^k + 1 is irreducible (mod 2), or -1 if no such k exists.
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2
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1, 1, 2, 1, 3, 2, 3, 1, 3, 5, 1, 3, 4, 6, -1, 1, 4, 5, 8, 3, 7, 2, 9, 3, 5, 7, 9, -1, 5, 9, 1, 4, 7, 8, 11, 14, -1, 3, 5, 6, 11, 12, 14, 3, 7, 9, 11, 15, -1, 3, 5, 15, 17, 2, 7, 14, 19, 1, 21, 5, 9, 14, 18, -1, 3, 7, 18, 22, -1, -1, 1, 3, 9, 13, 15, 19, 25, 27, 2, 27, 1, 9, 21, 29, 3, 6, 7, 13
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OFFSET
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2,3
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COMMENTS
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Row n (if it is not -1) is invariant under the map k -> n-k. - Robert Israel, Mar 14 2018
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REFERENCES
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Alanen, J. D., and Donald E. Knuth. "Tables of finite fields." Sankhyā: The Indian Journal of Statistics, Series A (1964): 305-328.
John Brillhart, On primitive trinomials (mod 2), unpublished Bell Labs Memorandum, 1968.
Marsh, Richard W. Table of irreducible polynomials over GF (2) through degree 19. Office of Technical Services, US Department of Commerce, 1957.
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LINKS
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EXAMPLE
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Triangle begins:
1,
1, 2,
1, 3,
2, 3,
1, 3, 5,
1, 3, 4, 6,
-1,
1, 4, 5, 8,
3, 7,
2, 9,
3, 5, 7, 9,
-1,
5, 9,
1, 4, 7, 8, 11, 14,
-1,
3, 5, 6, 11, 12, 14,
3, 7, 9, 11, 15,
-1,
3, 5, 15, 17,
2, 7, 14, 19,
1, 21,
...
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MAPLE
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for n from 2 to 30 do
S:= select(k -> Irreduc(x^n+x^k+1) mod 2, [$1..n-1]);
if S = [] then print(-1) else print(op(S)) fi
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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