Why is it that each aliyah for Krias Hatorah has to be 3 pesukim at minimum? I know that the minimum for an entire krias Hatorah on a weekday is 10 pesukim (with the exception of Purim being 9 pesukim), but why do the pesukim have to be split evenly amongst those called up?
The Talmud states that this is to correspond to the three-fold division of Torah, Neviim, and Ketuvim:
הני ג' פסוקין כנגד מי אמר רב אסי כנגד תורה נביאים וכתובים
What do these three verses represent? — R. Assi said: The Pentateuch, the Prophets, and the Hagiographa. (Soncino translation)
Most of the Codes simply cite the rule without mentioning the reason. Some of the commentaries to the Codes reference the Talmud's reasoning. However, R. Mordechai Yaffe in his Code provides a different reason. He simply says that the Sages assessed three verses to be the minimal amount that could be considered a significant reading for one person:
ואין קורין עם כל אחד פחות מג' פסוקים שכן שיערו רבותינו ז"ל ג"כ שזה סגי לקרות קריאה חשובה לאיש אחד