A205532 -id:A205532

Search: a205532 -id:a205532

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A205534

Record values of A205531 and A205535.

+10
5

1, 3, 5, 6, 9, 11, 15, 17, 21, 27, 29, 35, 39, 45, 51

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Related to the 4.X Selfridge Conjecture by P. Underwood, cf. link.

Records occur at [p=A205532(n), y=A205534(n)] = [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], [9257329, 51],...

It appears that all records > 6 occur at elements of A102295.

LINKS

Table of n, a(n) for n=1..15.

P. Underwood, 4.X Selfridge Conjecture (on "Prime Pages" profile), Jan 2012.

PROG

(PARI) m=-1; forprime(p=1, default(primelimit), if(m+0<m=max(m, A205535(p), m), print1(m", ")))

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 28 2012

STATUS

approved

A205531

Least nonnegative integer y such that Kronecker(y^2 - 4, p(n)) == -1 and (x+2)^(p(n)+1) == 5 -+ 2*y (mod p(n), mod x^2 +- y*x + 1).

+10
4

1, 0, 1, 0, 0, 3, 1, 0, 0, 1, 0, 3, 1, 0, 0, 1, 0, 5, 0, 0, 3, 0, 0, 1, 3, 1, 0, 0, 6, 1, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 0, 5, 0, 3, 1, 0, 0, 0, 0, 5, 1, 0, 5, 0, 1, 0, 1, 0, 3, 1, 0, 1, 0, 0, 3, 1, 0, 3, 0, 5, 1, 0, 0, 3, 0, 0, 1, 3, 1, 5, 0, 6, 0, 3, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 0, 1, 0, 1, 0, 3, 0, 1, 0, 5, 0, 3, 1, 0, 0, 1, 0, 0, 1, 0, 5, 3, 1, 0, 0, 1, 6, 0, 0, 3, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

Related to the 4.X Selfridge Conjecture by P. Underwood, which states that p is prime iff such a y exists.

Records occur at [p=prime(k),y=a(k)] = [A205532(n), A205534(n)] = [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ...

LINKS

Table of n, a(n) for n=1..132.

P. Underwood, 4.X Selfridge Conjecture (on "Prime Pages" profile), Jan 2012.

PROG

(PARI) A205531(n)=A205535(prime(n))

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 28 2012

STATUS

approved

A205526

Least positive integer y such that Kronecker(y^2 - 4, p(n)) == -1 and (x+2)^(p(n)+1) == 5 -+ 2*y (mod p(n), mod x^2 +- y*x + 1).

+10
1

1, 3, 1, 3, 1, 3, 1, 4, 1, 1, 4, 3, 1, 3, 1, 1, 1, 5, 3, 1, 3, 4, 1, 1, 3, 1, 3, 1, 6, 1, 3, 1, 1, 4, 1, 4, 3, 3, 1, 1, 1, 5, 1, 3, 1, 4, 4, 3, 1, 5, 1, 1, 5, 1, 1, 1, 1, 4, 3, 1, 3, 1, 3, 1, 3, 1, 4, 3, 1, 5, 1, 1, 3, 3, 4, 1, 1, 3, 1, 5, 1, 6, 1, 3, 4, 1, 1, 3, 1, 3, 1, 1, 3, 1, 4, 1, 1, 1, 3, 6, 3, 1, 1, 1, 4, 3, 1, 1, 1, 5, 3, 3, 1, 4, 4, 1, 3, 1, 1, 1, 5, 3, 1, 1, 4, 1, 6, 1, 3, 3, 4, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is an alternate version of A205531, which should be considered as the main entry. One has A205526(n)=A205531(n) whenever the latter is nonzero.

Related to the 4.X Selfridge Conjecture by P. Underwood, which states that p is prime iff such an y exists.

Records are [p, y] = [A205532(n), A205534(n)] = [2, 1], [3, 3], [19, 4], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ...

LINKS

Table of n, a(n) for n=1..132.

P. Underwood, 4.X Selfridge Conjecture (on "Prime Pages" profile), Jan 2012.

PROG

(PARI) a(n)={n=prime(n); for(y=1, 1e7, kronecker(y^2-4, n)==-1 | next;

Mod(x+Mod(2, n), x^2-y*x+1)^(n+1)==5+2*y | next; Mod(x+Mod(2, n), x^2+y*x+1)^(n+1)==5-2*y & return(y))}

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 28 2012

STATUS

approved

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