Is 1 a Prime Number?

Check out my series on building numbers from the ground up:

Okay, we all know 1 isn’t prime. But it wasn’t always that way. It also wasn’t always considered to be a number at all. Join me on a deep dive into mathematical history to see how our concepts of numbers and primes changed over time.

A huge thank you to my patrons for keeping the channel alive. Please consider supporting me by following the link above!

An enormous thank you to Chris Caldwell et al. for their compilation of sources regarding the history of 1 as a prime. Their work is much more thorough and extensive than my video and covers many, many more examples of 1 defined as prime or otherwise. My video is intended as a rough narrative through line regarding the broad(ish) consensus regarding the status of 1. Having such an extensive list of sources was really helpful when writing. You can find their article here:

Caldwell, Chris K. et al. “The History of the Primality of One: A Selection of Sources.” Journal of integer sequences 15.9 (2012).

All music by Danijel Zambo.

TIMESTAMPS:

00:00 - Intro
01:20 - Antiquity
09:17 - Medieval
13:50 - Modern
23:59 - Outro

REFERENCES

[1] - Euclid. The Thirteen Books of The Elements, Vol. 2 Books III-IX. Translated by Heiberg. Dover Books on Mathematics, 1956.

[2] - Martianus Capella. et al. Martianus Capella and the Seven Liberal Arts. Vol.2, The Marriage of Philology and Mercury. New York ;: Columbia University Press, 1977.

[3] - Isidore, and Stephen A. Barney. The Etymologies of Isidore of Seville. Cambridge: Cambridge University Press, 2006.

[5] - Stevin, S. The Principal Works of Simon Stevin. Vol.2B. Mathematics. C.V. Swets & Zeitlinger, 1958.

[6] - Morland, Samuel. The Description and Use of Two Arithmetick Instruments : Together with a Short Treatise, Explaining and Demonstrating the Ordinary Operations of Arithmetick, as Likewise a Perpetual Almanack and Several Useful Tables : Presented to His Most Excellent Majesty Charles II ... London: Printed and are to be sold by Moses Pitt ..., 1673.

[7] - Moxon, Joseph, and Henry Coley. Mathematicks Made Easie, or, A Mathematical Dictionary ... The second edition, corrected and much enlarged by Hen. Coley ... London: Printed for J. Moxon ..., 1692.

[8] - Rahn, Johann Heinrich, Thomas Brancker, and John Pell. An Introduction to Algebra. London: Printed by W.G. for Moses Pitt ..., 1668.

[9] - C. Goldbach, Letter to Euler dated 7 June 1742. Correspondance math´ematique et physique de quelques c´elebres geometres du XVIIIeme, siecle by P.-H. Fuss, Vol. I, pp. 125–129, Academie Imperiale des Sciences, St.-Petersbourg, 1843.

[10] - J. G. Krüger, Gedancken von der Algebra, nebst den Primzahlen von 1 bis 1000000, Lüderwalds Buchhandlung, Halle, 1746.

[11] - Euler, Leonhard, and John Hewlett. Elements of Algebra. Trans. John Hewlett. Cambridge: Cambridge University Press, 2009.

[12] - Gauss, Carl Friedrich et al. Disquisitiones Arithmeticae. New York: Springer-Verlag, 1986. Print.

[13] - Gregory, Olinthus. Mathematics for Practical Men : Being a Common-Place Book of Pure and Mixed Mathematics, Designed Chiefly for the Use of Civil Engineers, Architects and Surveyors. 4th ed / rev. and enl., by Henry Law. London: J. Weale, 1862.

[14] - Hardy, G. H. (Godfrey Harold). A Course of Pure Mathematics. 6th ed. Cambridge Eng: The University Press, 1933.

[15] - Hardy, G. H. (Godfrey Harold). A Course of Pure Mathematics. Centenary ed. Cambridge: Cambridge University Press, 2008.

[16] - Sagan, C. Contact. Simon and Schuster, 1985.

[17] - J. B. Andreasen, L.-A. T. Spalding, and E. Ortiz, FTCE: Elementary Education K-6, Cliffs Notes, Wiley, Hoboken, NJ, 2010.

[MarkAhlquist]

The number one should really be considered a planet.

[Reydriel]

I've always rationalised it even further by thinking that 1 is a sort of "superprime", which is sort of true because it is the multiplicative identity. We have to separate it from the normal primes because it screws with a lot of theorems involving prime numbers.

[caladbolg]

I'm glad you're not strictly talking about math problems but also about the history and philosophy behind how we define things. And a video on deductive reasoning? Everything I ever wished for from a math channel.

[pokerformuppets]

I seem to remember 1 becoming "un-prime" when I was about 10. As you're classifying that as "medieval", I now feel positively ancient!

[Roboman]

I love that you've kept up with the humor. It's really fun to watch your videos and I look forward to more!

[BobcatTrebuchet]

Another great video! I liked the bit about the Totient function as I hadn't come across it before!

[atzuras]

In the Contact movie/book it makes some sense to include '1' since the aliens were replying to early radio and TV emissions from 1936 when possibly the 1 was listed as a prime by some sources as you just shown.
Also, it makes sense to start a sequence with the fundamental unity of measurement in the same way that the
messages on the Voyager included distances based on a multiple (in base 2) of the fundamental frequency of the hydrogen atom.

[dizwell]

Thank you for resolving a 50+ year annoyance! I learned some maths before formal schooling from some of my dad's 1930s textbooks: 1 was listed as a prime.

Then I went to secondary school and got (literally!) knuckle-rapped for answering 1 was prime, because the "new maths" had decided 2 was the first prime.

The discrepancy has always irritated me ever since. Your explanation from Gauss finally makes me happy!

Nice video: thank you.

[zachrodan]

21:33 that was BY FAR the most convincing argument i have heard for demoting pluto from planethood.

[dougthayer]

I think what I find so appealing about your videos is that they tend to take ideas that a lot of mathsy people are probably familiar with (the representation of numbers as sets, the idea that one is not a prime, etc.) and shine a light on why we got there, rather than just the nuts and bolts of how those ideas work. I hope this isn't accidentally an insult, but I feel like the most enticing part of your videos has been that they straddle the border between mathematics and philosophy, albeit clearly on team mathematics. I think there has been some exploration of this space, but usually not with the level of sincerity and depth that you bring. Once again, an exceptional video!

[zarchblarch]

I feel like the totient function excluding 1 isn’t because 1 doesn’t work as a prime, but because it isn’t being counted as a factor. If you count all the numbers less than 20 that share a factor and you consider 1 to be a factor, then there are none because 1 is a factor of everything.

[pamdemonia]

I've been so looking forward to this, and you did not disappoint! Thank you for amazing math content that tickles my brain in a very wonderful way.

[ToriKo_]

Great video. I really liked how you used this topic to talk about definitions and properties more broadly. I like how even if you’re talking about a ‘simpler’ topic you try to find the conclusion that helps a student to learn about math more broadly

[Scum]

Oh man, "until next prime" is a great closing stinger. If you used it just as a one-off for this episode, I urge you to consider making it your standard closing line.

[chrisj.]

Another Roof, you have been selecting some awesome math topics! To create some connective tissue, I noticed the Euclidian definition of the unit and then the number as a collection of units sounds a lot like the foundation for the "empty set" definition of numbers you outlined in a previous video. I didn't realize that method of number definition had such old historical precedent...

Anyway, looking forward to future videos. Keep up the good work!

[menso]

I never knew Hector Salamance was into pure math

[lexinwonderland]

i'm so glad you mentioned the distinction of primes/units in general rings like the integers! That's what I was watching for the whole time lol, definitely wish you mentioned a bit more about it but fantastic video as always!

[soletrador]

Another perfect video! Congratulations!

[vortygames]

next topic will be about number theory, i see..

[asoulintodarkness]

It's a cool video!!! Im hoping to see other topics in your channel cuase you explain so well but I enjoyed this one as well!