👩🏼‍⚕️ 🤛🏻 🚋 Proth's theorem 🖤 💂 🏗️

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François Prot (1852-1879) was a self-taught farmer who lived in the French village of Vaudevan-Damloux near Verdun. The theorem considered here is one of four results he obtained that can be used to test the simplicity of numbers. It was published in the French scientific journal Comptes rendus de l'Académie des Sciences (Fig. 1) in 1878. Protus probably had proofs of his results, but he did not present them.

Definition

– . , , , . , , – .

– $k \ cdot2 ^ n + 1$ , – . , k <2 ^ n , . , 448 , $7 \ cdot2 ^ 6 + 1$ , 2 ^ 6> 7 .

: 3, 5, 9,13,17, 25, 33… - A080075.

, , : 3, 5, 13, 17, 41, 97, 113… - A080076.

$10223 \ cdot2 ^ {31172165} +1$ , 9,383,761 . , , 2 ^ n-1 . 31 2016 Seventeen or Bust $k \ cdot2 ^ n + 1$ .

, . , ( $n \ cdot2 ^ n + 1$ ) k = n , ( $2 ^ {2 ^ n} +1$ ) – k = 1 .

– , , :

$a ^ {\ frac {p-1} {2}} \ equiv-1 ~ (\ text {mod} ~ p)$

. , $x ^ 2 \ equiv a ~ (\ text {mod} ~ m)$ . , , .

, , . , .

m> 2 – . , , $a ^ {(p-1) / 2} \ equiv1 ~ (\ text {mod} ~ p)$ , $a ^ {(p-1) / 2} \ equiv-1 ~ (\ text {mod} ~ p)$ .

, – eleven , : $2 ^ 5 = 32 \ equiv-1 ~ (\ text {mod} ~ 11)$ . – eleven $3 ^ 5 = 243 \ equiv1 ~ (\ text {mod} ~ 11)$ .

. , . , .

– n-1 , $q> \ sqrt {n} -1$ . , $a ^ {n-1} \ equiv1 ~ (\ text {mod} ~ n)$ $a ^ {(n-1) / q} -1$ , – .

. n = 7 . , . n-1 , . , . q = 3 . $3> \ sqrt {7} -1 \ approx1.65$ . , . a = 2 , $2 ^ {6} \ equiv1 ~ (\ text {mod} ~ 7)$ $2 ^ {(7-1) / 3} -1 = 3$ . , .

, one . , , k> 1 . : $q ^ k> \ sqrt {n} -1$ .

, n = 17 . n-1 = 16 q = 2 . , $2> \ sqrt {17} -1 \ approx3,12$ . q = 2 k = 4 . : $2 ^ 4> \ sqrt {17} -1 \ approx3,12$ . , , , , : n = 17 – .

. , $a ^ {(p-1) / 2} \ equiv-1 ~ (\ text {mod} ~ p)$ .

– . $a ^ {(p-1) / 2} \ equiv-1 ~ (\ text {mod} ~ p)$ . .

, , $a ^ {(p-1) / 2} \ equiv-1 ~ (\ text {mod} ~ p)$ .

$a ^ {(p-1) / 2} \ equiv-1 ~ (\ text {mod} ~ p)$ .

n = p = 2 ^ k + 1 . q = 2 n-1 .

, :

$a ^ {n-1} = \ left (a ^ {(n-1) / 2} \ right) ^ 2 \ equiv1 ~ (\ text {mod} ~ n)$ –
$a ^ {(n-1) / q} -1$ –

$2 ^ k> \ sqrt {n} -1$ . n = p . .

. , , . , , , . .

, . , $a \ not \ equiv1 ~ (\ text {mod} ~ N)$ . $b = a ^ {(n-1) / 2}$ .

$b \ equiv-1 ~ (\ text {mod} ~ N)$ . , – .
$b \ not \ equiv \ pm1 ~ (\ text {mod} ~ N)$ $b ^ 2 \ equiv1 ~ (\ text {mod} ~ N)$ . , , – , ( $b \ pm1, N$ ) .
$b ^ 2 \ not \ equiv1 ~ (\ text {mod} ~ N)$ . – .
$b \ equiv1 ~ (\ text {mod} ~ N)$ . .

, . , 1/2 .

, . , . , 1,2, ..., N-1 (N-1) / 2 . , 1/2 . , – .

. N = 17 . . , a = 2 , b = 256 . . , $256 \ equiv1 ~ (\ text {mod} ~ 17)$ . . , .

a = 3 . b = 6561 $6561 \ equiv-1 ~ (\ text {mod} ~ 17)$ . , , N = 17 .

, – , , . , . «» . .

N = 9 , . . b = 16 . N = 9 . , $16 \ equiv7 \ neq \ pm1 ~ (\ text {mod} ~ 9)$ . . , N = 9 .

-. , . , « », « ». , , . .

2008 , , $O ((k \ log k + \ log N) (\ log N) ^ 2)$ . "Proth.exe", .

N = r ^ et + 1 , – , – , $e, t \ geq1$ . r ^ e> t . , $a ^ {N-1} \ equiv1 ~ (\ text {mod} ~ N)$ $a ^ {(N-1) / r} \ not \ equiv1 ~ (\ text {mod} ~ N)$ , – .

. , N = 17 . $N = 17 = 2 ^ 4 \ cdot1 + 1$ . r = 2, ~ e = 4, ~ t = 1 . , . . a = 3 . :

$3 ^ {16} \ equiv1 ~ (\ text {mod} ~ 17)$ –
$3 ^ {8} \ not \ equiv1 ~ (\ text {mod} ~ 17)$ –

, , N = 17 .

Of course, Proth's generalized theorem is applicable for a large number of number groups, but the selection of the necessary variables is too time-consuming. Therefore, in practice, the classical theorem is used much more often.

Proth's theorem

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