Numeri Idonei (con­ve­nient)

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Numeri idonei
What are these numeri idonei of Euler? Also called con­ve­nient
num­bers, they were used con­ve­niently by Euler to pro­duce prime
num­bers.
Now I will explain what the numeri idonei are. Let n ≥ 1. If q is an
odd prime and there exist inte­gers x, y ≥ 0 such that q = x² + ny² ‚
then:
(i) gcd(x, ny) = 1;
(ii) if q = x₁ ² + ny₁² with inte­gers x 1 , y 1 ≥ 0, then x = x 1 and y = y 1 .
We may ask the fol­low­ing ques­tion. Assume that q is an odd in–
teger, and that q = x ² + ny ² , with inte­gers x, y ≥ 0, such that
con­di­tions (i) and (ii) above are sat­is­fied. Is q a prime num­ber?
The answer depends on n. If n = 1, the answer is “yes”, as Fer–
mat knew. For n = 11, the answer is “no”: 15 = 2 2 + 11 · 1 2 and
con­di­tions (i) and (ii) hold, but 15 is com­pos­ite. Euler called n a
numerus idoneus if the answer to the above ques­tion is “yes”.
Euler gave a cri­te­rion to ver­ify in a finite num­ber of steps whether
a given num­ber is con­ve­nient, but his proof was flawed. Later, in
1874, Grube found the fol­low­ing cri­te­rion, using in his proof results
of Gauss, which I will men­tion soon. Thus, n is a con­ve­nient num­ber
if and only if for every x ≥ 0 such that q = n + x ² ≤ 4n/​3 , if q = rs
and 2x ≤ r ≤ s, then r = s or r = 2x.
For exam­ple, 60 is a con­ve­nient num­ber, because
60 + 1 ² = 61 (),
60 + 2 ² = 64 = 4 · 16 = 8 · 8,
60 + 3 ² = 69 (),
60 + 4 ² = 76 ()
and the num­bers marked with a () do not have a fac­tor­iza­tion of
the form indi­cated.
Euler showed, for exam­ple, that 1848 is a con­ve­nient num­ber,
and that
q = 18518809 = 197 ² + 1848 · 100 ²
is a prime num­ber. At Euler’s time, this was quite a feat.
Gauss under­stood con­ve­nient num­bers in terms of his the­ory of
binary qua­dratic forms. The num­ber n is con­ve­nient if and only if
each genus of the form x ² + ny ² has only one class.
Here is a list of the 65 con­ve­nient num­bers found by Euler:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25,
28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93,
102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240,
253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840,
1320, 1365, 1848.
Are there other con­ve­nient num­bers? Chowla showed that there
are only finitely many con­ve­nient num­bers; later, finer ana­lyt­i­cal
work (for exam­ple, by Briggs, Gross­wald, and Wein­berger)
implied that there are at most 66 con­ve­nient num­bers.
The prob­lem is dif­fi­cult. The exclu­sion of an addi­tional numerus
idoneus is of a kind sim­i­lar to the exclu­sion of a hypo­thet­i­cal tenth358
imag­i­nary qua­dratic field (by Heeg­ner, Stark, and Baker), which
I have already mentioned.

My num­bers, my friends /​Paulo Ribenboim

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