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 A Perplexing (Prime) Puzzle (Posted on 2007-06-05)
Consider the quadruplets (p,q,r,s) of positive integers with p>q>r>s, and satisfying pr+qs= (q+s+p-r)(q+s-p+r).

Is it ever the case that pq+rs is a prime number?

 See The Solution Submitted by K Sengupta Rating: 4.0000 (3 votes)

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 Some number crunching | Comment 1 of 5

I put together a small program to find sets of p,q,r,s which satisfy p>q>r>s>0, p*r+q*s = (q+s+p-r)(q+s-p+r), and gcd(p*q, r*s) = 1.

The values v = p*q+r*s which came out for 50>=p are below.  There were 347 values with 100>=p and there were 15803 values with 500>=p.

v=104 p=11 q=9 r=5 s=1
v=147 p=13 q=11 r=4 s=1
v=152 p=13 q=11 r=9 s=1
v=245 p=17 q=13 r=8 s=3
v=248 p=17 q=13 r=9 s=3
v=315 p=19 q=16 r=11 s=1
v=441 p=23 q=17 r=10 s=5
v=456 p=23 q=17 r=13 s=5
v=455 p=23 q=19 r=6 s=3
v=455 p=23 q=19 r=9 s=2
v=465 p=23 q=19 r=14 s=2
v=488 p=23 q=19 r=17 s=3
v=539 p=25 q=20 r=13 s=3
v=632 p=27 q=23 r=11 s=1
v=637 p=27 q=23 r=16 s=1
v=715 p=29 q=23 r=12 s=4
v=735 p=29 q=23 r=17 s=4
v=744 p=29 q=25 r=19 s=1
v=777 p=29 q=26 r=23 s=1
v=819 p=31 q=25 r=11 s=4
v=833 p=31 q=26 r=9 s=3
v=845 p=32 q=24 r=11 s=7
v=931 p=33 q=27 r=8 s=5
v=1016 p=33 q=27 r=25 s=5
v=1001 p=34 q=29 r=15 s=1
v=1005 p=34 q=29 r=19 s=1
v=1183 p=37 q=31 r=18 s=2
v=1185 p=37 q=31 r=19 s=2
v=1197 p=37 q=32 r=13 s=1
v=1235 p=37 q=33 r=7 s=2
v=1240 p=39 q=27 r=17 s=11
v=1295 p=39 q=27 r=22 s=11
v=1304 p=39 q=31 r=19 s=5
v=1309 p=39 q=31 r=20 s=5
v=1365 p=41 q=29 r=16 s=11
v=1464 p=41 q=29 r=25 s=11
v=1407 p=41 q=31 r=17 s=8
v=1463 p=41 q=31 r=24 s=8
v=1449 p=41 q=34 r=11 s=5
v=1463 p=41 q=34 r=23 s=3
v=1505 p=41 q=36 r=29 s=1
v=1520 p=43 q=29 r=21 s=13
v=1533 p=43 q=29 r=22 s=13
v=1547 p=43 q=32 r=19 s=9
v=1547 p=43 q=33 r=16 s=8
v=1665 p=43 q=34 r=29 s=7
v=1608 p=43 q=37 r=17 s=1
v=1617 p=43 q=37 r=26 s=1
v=1729 p=44 q=36 r=29 s=5
v=1771 p=45 q=39 r=16 s=1
v=1784 p=45 q=39 r=29 s=1
v=1729 p=46 q=34 r=15 s=11
v=1905 p=46 q=34 r=31 s=11
v=1928 p=47 q=33 r=29 s=13
v=1859 p=47 q=37 r=15 s=8
v=1872 p=47 q=37 r=19 s=7
v=1935 p=47 q=37 r=28 s=7
v=1995 p=47 q=37 r=32 s=8
v=1911 p=47 q=38 r=25 s=5
v=1960 p=47 q=41 r=11 s=3
v=1953 p=47 q=41 r=13 s=2
v=1995 p=47 q=41 r=34 s=2
v=2035 p=47 q=41 r=36 s=3
v=2261 p=50 q=45 r=11 s=1

 Posted by Brian Smith on 2007-06-07 00:34:43

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