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A269598
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Irregular triangle giving T(n, k) = -(2*A269596(n, k))^(prime(n)-2) modulo prime(n) for n >= 2.
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1
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1, 1, 2, 5, 1, 3, 9, 4, 8, 1, 5, 9, 1, 2, 3, 8, 6, 2, 6, 1, 14, 5, 4, 7, 8, 11, 7, 4, 3, 13, 17, 1, 14, 9, 10, 20, 21, 14, 19, 8, 1, 17, 18, 16, 11, 6, 26, 10, 8, 1, 2, 18, 13, 24, 12, 9, 25, 7, 14
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OFFSET
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2,3
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COMMENTS
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The length of row n >= 2 is (prime(n)-1)/2 = A005097(n-1).
The irregular companion triangle -(2*A269597(n, k))^(prime(n)-2) modulo prime(n) is given in A269599.
These numbers, called z_1 = z_1(x_1,prime(n)), appear in a recurrence for the approximation sequence {x_n(prime(n), b, x1)} of the p-adic integer sqrt(-b) with entries congruent to x1 modulo prime(n). The irregular triangle for the b values is given in A269595(n, k) for n >= 2 (odd primes), and A269596(n, k) gives the corresponding x1 values.
T(n, k) is the unique solution of the first order congruence 2*A269596(n, k)*z(n, k) + 1 == 0 (mod prime(n)), with 0 <= z(n, k) <= prime(n)-1, for n >= 2.
For a(n), n >= 2, see column z_1 of the table of the paper given as a Wolfdieter Lang link.
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LINKS
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FORMULA
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T(n, k) = modp(-(2*A269596(n, k))^(prime(n) -2), prime(n)), for n >= 2 and k=1, 2, ...., (prime(n)-1)/2, with modp(a, p) giving the number a' from {0, 1, ..., p-1} with a' == a (mod p).
T(n, k) = prime(n) - A269599(n, k).
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EXAMPLE
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The irregular triangle T(n, k) begins (P(n) stands here for prime(n)):
n, P(n)\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2, 3: 1
3, 5: 1 2
4, 7: 5 1 3
5, 11: 9 4 8 1 5
6: 13: 9 1 2 3 8 6
7, 17: 2 6 1 14 5 4 7 8
8, 19: 11 7 4 3 13 17 1 14 9
9, 23: 10 20 21 14 19 8 1 17 18 16 11
10, 29: 6 26 10 8 1 2 18 13 24 12 9 25 7 14
...
T(5, 3) = 8 because 2*A269596(5, 3)*8 + 1 = 2*2*8 + 1 = 33 == 0 mod 11, hence modp(33, 11) = 0 , and 8 is the unique nonnegative solution <= 10 of 2*A269596(5, 3)*z + 1 == 0 (mod 11).
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MATHEMATICA
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nn = 12; s = Table[Select[Range[Prime@ n - 1], JacobiSymbol[#, Prime@ n] == 1 &], {n, nn}]; t = Table[Prime@ n - s[[n, (Prime@ n - 1)/2 - k + 1]], {n, Length@ s}, {k, (Prime@ n - 1)/2}] /. {} -> {1}; u = Prepend[Table[SelectFirst[Range@ #, Function[x, Mod[x^2 + t[[n, k]], #] == 0]] &@ Prime@ n, {n, 2, Length@ t}, {k, (Prime@ n - 1)/2}], {1}]; Table[SelectFirst[Range@ #, Function[z, Mod[-(2 u[[n, k]] z + 1), #] == 0]] &@ Prime@ n, {n, 2, Length@ u}, {k, (Prime@ n - 1)/2}] // Flatten (* Michael De Vlieger, Apr 04 2016, Version 10 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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