OFFSET
1,1
COMMENTS
This sequence has a restriction involving 4 variables. More composite cases are described with a better restrictive expression. The expression for k(a,c,d,z) will force k^2 + k + 41 to be either a fraction or a composite number.
The condition on k(a,c,d,z) was determined by quadratic curve fitting. It has been automated with the Maple Interactive() command. The ultimate motivation is to try to find a closed-from expression that generates all the composite cases of k^2 + k + 41 for integer k.
REFERENCES
John Stillwell, Elements of Number Theory, Springer, 2003, page 3.
LINKS
Matt C. Anderson, Graph of composite values for n^2 + n + 41 with a modular symmetry.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
MAPLE
# Euler considered the prime values for n^2 + n + 41;
# This is a 76 second calculation on a 2.93 GHz machine
h := n^2+n+41;
y := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c;
y2 := subs(n = y, h);
y3 := factor(y2);
# note that y is an expression in 4 variables.
# After a composition of functions, an algebraic factorization
# can be observed in y3. As long as y3 is an integer, it will
# be composite. This is because y3 factors and both factors
# are integers bigger than one.
maxn := 6000;
A := {}:
for n to maxn do
g := n^2+n+41:
if isprime(g) = false then A := `union`(A, {n}) end if :
end do:
# now the A set contains composite values of the form
# n^2 + n + 41 less than maxn.
c := 1: a := 1: d := 1: z := -1: p := 41:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c:
A2 := A:
while q < maxn do
while `and`(q < maxn, d < 100) do
while q < maxn do while
q < maxn do
A2 := `minus`(A2, {q});
A2 := `minus`(A2, {c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c});
z := z+1;
A2 := `minus`(A2, {c*a*z^2-((((d+2)*(1/3))*c-2)*a/d+1)*(1*z)+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c}); q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c
end do;
a := a+1; z := -1;
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do;
d := d+1: a := 1:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do:
c := c+1: d := 1:
q := c*a*z^2+((((d+2)*(1/3))*c-2)*a/d+1)*z+((((367*d^2+d+1)*(1/9))*c^2-((d+2)*(1/3))*c+1)*a/d^2-(((d-1)*(1/3))*c+1)/d)/c :
end do:
A2;
# Matt C. Anderson, May 13 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Matt C. Anderson, Apr 27 2014
STATUS
approved