How to create an infinite Table of numbers

You can do something like this

n = 70;
Do[If[5^a + 5^b == 5^c + 5^d, Print[{a, b, c, d}]], {a, 0, n}, {b, 
  a + 1, n}, {c, b + 1, n}, {d, c + 1, n}]

of course, $70llinfty$ but I am quite confident that the only solutions are $$a=c, b=da=d, b=c.$$

Say $a=pm 5^{n_a},b=pm 5^{n_b},c=pm 5^{n_c}$ and $d=pm 5^{n_d}$ and that WLOG $n_a$ is the smallest. Then $$5^{n_b-n_a}pm 1=pm 5^{n_c-n_a}pm 5^{n_d-n_a}.$$

The right hand side is divisible by 5 unless $n_a=n_d$ or $n_a=n_c$, and the LHS is not, unless $n_a=n_b$. So the only solutions are

$$(a,b,c,d)=(pm 5^{n_1},mp 5^{n_1},pm 5^{n_2},mp5^{n_2}),$$ or $$(a,b,c,d)=(pm 5^{n_1},pm 5^{n_2},pm 5^{n_2},pm 5^{n_1}),$$ or $$(a,b,c,d)=(pm 5^{n_1},pm 5^{n_2},pm 5^{n_1},pm 5^{n_2}).$$