In the following code, I used two \cmidrules from the booktabs package in combination with a horizontal white space in order to visually separate the two pairs of columns without using a vertical line. I also used the makecell package in order to add a linebreak in the column headers. Additionally, I added siunitx in order to improve the alignment of the numbers in their columns.

\documentclass{article}
\usepackage[greek]{babel}
\usepackage{booktabs}
\usepackage{siunitx}
\usepackage{makecell}
\renewcommand{\theadfont}{\normalsize}
\begin{document}
\begin{table}[htp]
\belowcaptionskip 0.5ex
\centering
\begin{tabular}{S[table-format=2]S[table-format=1]@{\qquad}S[table-format=3]S[table-format=2]}
\toprule
{\thead{Πρώτος}} & {\thead{Ελάχιστη θετική\ πρωταρχική ρίζα}}
& {\thead{Πρώτος}} & {\thead{Ελάχιστη θετική\ πρωταρχική ρίζα}} \
\cmidrule(r{2em}){1-2} \cmidrule(l){3-4}
2 & 1 & 89 & 3 \
3 & 2 & 97 & 5 \
5 & 2 & 101 & 2 \
7 & 3 & 103 & 5 \
11 & 2 & 107 & 2 \ \addlinespace
13 & 2 & 109 & 6 \
17 & 3 & 113 & 3 \
19 & 2 & 127 & 3 \
23 & 5 & 131 & 2 \
29 & 2 & 137 & 3 \ \addlinespace
31 & 3 & 139 & 2 \
37 & 2 & 149 & 2 \
41 & 6 & 151 & 6 \
43 & 3 & 157 & 5 \
47 & 5 & 163 & 2 \ \addlinespace
53 & 2 & 167 & 5 \
59 & 2 & 173 & 2 \
61 & 2 & 179 & 2 \
67 & 2 & 181 & 2 \
71 & 7 & 191 & 19 \ \addlinespace
73 & 5 & 193 & 5 \
79 & 3 & 197 & 2 \
83 & 2 & 199 & 3 \
\bottomrule
\end{tabular}
\caption{Οι ελάχιστες θετικές πρωταρχικές ρίζες των πρώτων < 200}
\label{Table:LeastPositivePrimitiveRoots}
\end{table}
\end{document}
To save some more space, you could also use three insted of two pairs of columns. With a slightly decreased font size in the column headers the whole table still fits into the textwidth while at the same only needing 16 instead of 23 rows:

\documentclass{article}
\usepackage[greek]{babel}
\usepackage{booktabs}
\usepackage{siunitx}
\usepackage{makecell}
\begin{document}
\begin{table}[htp]
\belowcaptionskip 0.5ex
\centering
\begin{tabular}{@{}S[table-format=2]S[table-format=1]@{\qquad}
S[table-format=3]S[table-format=1] @{\qquad}
S[table-format=3]S[table-format=2]@{}}
\toprule
{\thead{Πρώτος}} & {\thead{Ελάχιστη θετική\ πρωταρχική ρίζα}}
& {\thead{Πρώτος}} & {\thead{Ελάχιστη θετική\ πρωταρχική ρίζα}} & {\thead{Πρώτος}} & {\thead{Ελάχιστη θετική\ πρωταρχική ρίζα}} \
\cmidrule(r{2em}){1-2} \cmidrule(lr{2em}){3-4} \cmidrule(l){5-6}
2 & 1 & 59 & 2 & 137 & 3 \
3 & 2 & 61 & 2 & 139 & 2 \
5 & 2 & 67 & 2 & 149 & 2 \
7 & 3 & 71 & 7 & 151 & 6 \ \addlinespace
11 & 2 & 73 & 5 & 157 & 5 \
13 & 2 & 79 & 3 & 163 & 2 \
17 & 3 & 83 & 2 & 167 & 5 \
19 & 2 & 89 & 3 & 173 & 2 \ \addlinespace
23 & 5 & 97 & 5 & 179 & 2 \
29 & 2 & 101 & 2 & 181 & 2 \
31 & 3 & 103 & 5 & 191 & 19 \
37 & 2 & 107 & 2 & 193 & 5 \ \addlinespace
41 & 6 & 109 & 6 & 197 & 2 \
43 & 3 & 113 & 3 & 199 & 3 \
47 & 5 & 127 & 3 \
53 & 2 & 131 & 2 \
\bottomrule
\end{tabular}
\caption{Οι ελάχιστες θετικές πρωταρχικές ρίζες των πρώτων < 200}
\label{Table:LeastPositivePrimitiveRoots}
\end{table}
\end{document}
booktabs's rules. – chsk Apr 09 '21 at 14:42