Let
p
≥
7
p\geq 7
be a prime and
n
>
1
n>1
be a natural number. We show that there exist infinitely many Galois representations
ϱ
:
Gal
(
Q
¯
/
Q
)
→
GL
n
(
Z
p
)
\varrho :\operatorname {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})\rightarrow \operatorname {GL}_{n}(\mathbb {Z}_p)
which are unramified outside
{
p
,
∞
}
\{p, \infty \}
with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-
p
t
p^t
reduction map
SL
n
(
Z
p
)
→
SL
n
(
Z
/
p
t
Z
)
\operatorname {SL}_n(\mathbb {Z}_p)\rightarrow \operatorname {SL}_n(\mathbb {Z}/p^t\mathbb {Z})
, where
t
≔
8
(
n
2
−
n
)
(
3
+
⌊
log
p
(
2
n
+
1
)
⌋
)
+
8
t≔8(n^2-n)\left (3+\lfloor \operatorname {log}_p(2^n+1)\rfloor \right )+8
. The results are proven via a purely Galois theoretic lifting construction. When
p
≡
1
mod
4
p\equiv 1\mod {4}
, our results are conditional since in this case, we assume a very weak version of Vandiver’s conjecture.