Yeah sorry about that mix-up. We can do

In order to have an integer, (ad + bc) must equal 0. There is not much point letting a = 0 because then we would end up with a multiple of 5 which we already know is composite. Likewise there's no use letting b = 0 because then we have either a = 0 or d = 0, and if d = 0 then we are just multiplying two integers.

While I'm sure we can analyse this further, I took the easy way out (in my opinion) and wrote a brute force program. This is in Python, should run on any version

Code:

import math
def isprime(n):
if n < 2:
return False
# just trial division
if n%2 == 0:
return False
for i in range(3, int(math.floor(math.sqrt(n))+1), 2):
if n%i == 0:
return False
return True
for a in range(1,100):
for b in range(-75,75):
if b != 0:
for c in range(-75,75):
if (b*c) % a == 0:
d = -1*b*c / a
n = a*c+5*b*d
if n > 0 and isprime(n):
print "!"+str(n),a,b,c,d

The code isn't written to be elegant, just to require as little thought as possible to code.

A snippet of the output:

Code:

!89 6 -5 -6 -5
!3 6 -3 -2 -1
!31 6 -1 6 1
!31 6 1 6 -1
!3 6 3 -2 1
!89 6 5 -6 5

This means for example we can get product 3 by letting (a,b,c,d) = (6,-3,-2,-1). I added the exclamation marks just to make it easier to search.

So it looks like the first irreducible is 2, and the next irreducible isn't until 53!