On the System of Diophantine Equations 𝑥²−6𝑦²=−5 and 𝑥=𝑎𝑧²−𝑏

Table 2

Positive 𝑧-value of the solution.

	𝑎=1	𝑎=2	𝑎=3	𝑎=4	𝑎=5	𝑎=6	𝑎=7	𝑎=8	𝑎=9	𝑎=10
𝑏=1	𝑧=0	𝑧=0,1,2,3,6,91	𝑧=0	𝑧=0	𝑧=0	𝑧=0	𝑧=0	𝑧=0,1,3	𝑧=0	𝑧=0
𝑏=2	𝑧=1,3	NS	𝑧=1	NS	NS	NS	NS	NS	𝑧=1	NS
𝑏=3	𝑧=2	𝑧=1,59	NS	𝑧=1	𝑧=2	NS	NS	NS	NS	𝑧=1,83
𝑏=4	NS	NS	𝑧=1,5	NS	𝑧=1	NS	NS	NS	NS	NS
𝑏=5	𝑧=2	NS	𝑧=2	𝑧=1	NS	𝑧=1	NS	NS	NS	NS
𝑏=6	NS	NS	NS	NS	𝑧=1	NS	𝑧=1,5	NS	NS	NS
𝑏=7	𝑧=0	𝑧=0,2	𝑧=0	𝑧=0	𝑧=0	𝑧=0,1,2	𝑧=0	𝑧=0,1	𝑧=0	𝑧=0
𝑏=8	𝑧=1,3,5,41	NS	NS	NS	NS	NS	𝑧=1	NS	𝑧=1	NS
𝑏=9	𝑧=4	𝑧=1,2,29	NS	𝑧=2	𝑧=4	NS	NS	𝑧=1	NS	𝑧=1
𝑏=10	𝑧=3,9	NS	𝑧=1,3	NS	NS	NS	NS	NS	𝑧=1,3	NS

“NS” refers to “no solution.”