In fact red isn't a primary colour at all.
Tell that to your eyes, which have photoreceptors that respond to frequencies of EM radiation that we perceive as red, blue, and green light.
Actually, the primary colors we were taught in school are incorrect. When doing additive mixing, the primary colors are indeed red, green and blue. However, when you're mixing paint or ink, you're actually doing subtractive mixing, so the primary colors are the inversions of those three.
red = cyan
green = yellow
blue = magenta
You can reproduce the entire color spectrum from cyan, magenta, and yellow pigments. You cannot do so with red and blue pigments.
So red is indeed a primary color, as are blue, green, cyan, magenta, and yellow.