#### Title

Demonstratio theorematis Fermatiani omnem numerum primum formae 4*n*+1 esse summam duorum quadratorum

#### English Title

Proof of a theorem of Fermat that every prime number of the form 4*n*+1 is the sum of two squares

#### Enestrom Number

241

#### Fuss Index

60

#### Original Language

Latin

#### Content Summary

Euler probably already knew the "descent" step, as described in David Cox's *Primes of the Form x ^{2}+ny^{2}*. Here, he does the "reciprocity" step by using the Euler-Fermat theorem to say that

*a*

^{4n}−b

^{4n}is divisible by 4

*n*+1. Then he factors it as (

*a*

^{2n}+b

^{2n})(

*a*

^{2n}−b

^{2n}), and says that the second factor cannot be divisible by 4

*n*+1; hence the first factor is divisible by 4

*n*+1. As indexed by Eneström, this paper concludes on p. 13 of the

*Novi Commentarii*, but Euler's work continues immediately into E242.

#### Published as

Journal article

#### Published Date

1760

#### Written Date

1750

#### Original Source Citation

Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 3-13.

#### Opera Omnia Citation

Series 1, Volume 2, pp.328-337.

#### Record Created

2018-09-25