Neukirch (1999), §I.10, Exercise 1. Neukirch, Jürgen (1999), Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Berlin: Springer-Verlag, ISBN3-540-65399-6, MR1697859, Zbl0956.11021.
Neukirch (1999), §I.10, Exercise 1. Neukirch, Jürgen (1999), Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Berlin: Springer-Verlag, ISBN3-540-65399-6, MR1697859, Zbl0956.11021.
Sandifer, C. Edward, The Early Mathematics of Leonhard Euler (Washington, D.C.: The Mathematical Association of America, 2007), p. 253.
Leonhard Euler, "De summa seriei ex numeris primis formatae 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc. ubi numeri primi formae 4n − 1 habent signum positivum, formae autem 4n + 1 signum negativum" (On the sum of series [composed] of prime numbers arranged 1/3 − 1/5 + 1/7 + 1/11 − 1/13 − 1/17 + 1/19 + 1/23 − 1/29 + 1/31 etc., where the prime numbers of the form 4n − 1 have a positive sign, whereas [those] of the form 4n + 1 [have] a negative sign.) in: Leonhard Euler, Opuscula analytica (St. Petersburg, Russia: Imperial Academy of Sciences, 1785), vol. 2, pp. 240–256; see p. 241. From p. 241: "Quoniam porro numeri primi praeter binarium quasi a natura in duas classes distinguuntur, prouti fuerint vel formae 4n + 1, vel formae 4n − 1, dum priores omnes sunt summae duorum quadratorum, posteriores vero ab hac proprietate penitus excluduntur: series reciprocae ex utraque classes formatae, scillicet:1/5 + 1/13 + 1/17 + 1/29 + etc. et1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc.ambae erunt pariter infinitae, id quod etiam de omnibus speciebus numerorum primorum est tenendum. Ita si ex numeris primis ii tantum excerpantur, qui sunt formae 100n + 1, cuiusmodi sunt 101, 401, 601, 701, etc., non solum multitudo eorum est infinita, sed etiam summa huius seriei ex illis formatae, scillicet:1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc.etiam est infinita." (Since, further, prime numbers larger than two are divided as if by Nature into two classes, according as they were either of the form 4n + 1, or of the form 4n − 1, as all of the first are sums of two squares, but the latter are thoroughly excluded from this property: reciprocal series formed from both classes, namely:
1/5 + 1/13 + 1/17 + 1/29 + etc. and
1/3 + 1/7 + 1/11 + 1/19 + 1/23 + etc.
will both be equally infinite, which [property] likewise is to be had from all types of prime numbers. Thus, if there be chosen from the prime numbers only those that are of the form 100n + 1, of which kind are 101, 401, 601, 701, etc., not only the set of these is infinite, but likewise the sum of the series formed from that [set], namely:
1/101 + 1/401 + 1/601 + 1/701 + 1/1201 + 1/1301 + 1/1601 + 1/1801 + 1/1901 + etc.
likewise is infinite.)
doi.org
Shiu, D. K. L. (2000). "Strings of congruent primes". J. London Math. Soc. 61 (2): 359–373. doi:10.1112/s0024610799007863.
hathitrust.org
babel.hathitrust.org
Carl Friedrich Gauss, Disquisitiones arithmeticae (Leipzig, (Germany): Gerhard Fleischer, Jr., 1801), Section 297, pp. 507–508. From pp. 507–508: "Ill. Le Gendre ipse fatetur, demonstrationem theorematis, sub tali forma kt + l, designantibus k, l numeros inter se primos datos, t indefinitum, certo contineri numeros primos, satis difficilem videri, methodumque obiter addigitat, quae forsan illuc conducere possit; multae vero disquisitiones praeliminares necessariae nobis videntur, antequam hacce quidem via ad demonstrationem rigorosam pervenire liceat." (The illustrious Le Gendre himself admits [that] the proof of the theorem — [namely, that] among [integers of] the form kt + l, [where] k and l denote given integers [that are] prime among themselves [i.e., coprime] [and] t denotes a variable, surely prime numbers are contained — seems difficult enough, and incidentally, he points out a method that could perhaps lead to it; however, many preliminary and necessary investigations are [fore]seen by us before this [conjecture] may indeed reach the path to a rigorous proof.)
zbmath.org
Neukirch (1999), §I.10, Exercise 1. Neukirch, Jürgen (1999), Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Berlin: Springer-Verlag, ISBN3-540-65399-6, MR1697859, Zbl0956.11021.
Neukirch (1999), §I.10, Exercise 1. Neukirch, Jürgen (1999), Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Berlin: Springer-Verlag, ISBN3-540-65399-6, MR1697859, Zbl0956.11021.
