Does knowing how $5280$ is a product of primes help you to factor $5281$ as a product of primes?
The above question was mentioned in Abstract Algebra book by Gallian.
Prime factorization of
$5280 = 2^5 .3.5.11$
$5281 = 5281$
Does knowing how $5280$ is a product of primes help you to factor $5281$ as a product of primes?
The above question was mentioned in Abstract Algebra book by Gallian.
Prime factorization of
$5280 = 2^5 .3.5.11$
$5281 = 5281$
If there is suspicion that the number is prime and you have a complete prime factorization of $n-1,$ you may prove primality with the original Lucas primality test. I just programmed it in C++ ; the last line provides a proof that 5281 is prime, as all four powermod numbers are different from $1.$ Note that $-1 \equiv n-1 \pmod n$ is allowed and different from $1,$ we see $5280$ in the successful line.
Wed Sep 30 08:12:52 PDT 2020
5281 n-1: 5280 = 2^5 3 5 11
2 3 5 11
a: 2 q: 2 pmd: 1 q: 3 pmd: 3877 q: 5 pmd: 4166 q: 11 pmd: 845
a: 3 q: 2 pmd: 1 q: 3 pmd: 3877 q: 5 pmd: 4166 q: 11 pmd: 5156
a: 4 q: 2 pmd: 1 q: 3 pmd: 1403 q: 5 pmd: 2190 q: 11 pmd: 1090
a: 5 q: 2 pmd: 1 q: 3 pmd: 1 q: 5 pmd: 1 q: 11 pmd: 1090
a: 6 q: 2 pmd: 1 q: 3 pmd: 1403 q: 5 pmd: 2190 q: 11 pmd: 5276
a: 7 q: 2 pmd: 5280 q: 3 pmd: 1403 q: 5 pmd: 2190 q: 11 pmd: 5156
Here is a proof that Dietrich's example 1225723 is prime
n: 1225723 n-1: 1225722 = 2 3 281 727
2 3 281 727
a: 2 q: 2 pmd: 1225722 q: 3 pmd: 1 q: 281 pmd: 736434 q: 727 pmd: 605163
a: 3 q: 2 pmd: 1225722 q: 3 pmd: 1 q: 281 pmd: 956738 q: 727 pmd: 343840
a: 4 q: 2 pmd: 1 q: 3 pmd: 1 q: 281 pmd: 412053 q: 727 pmd: 738629
a: 5 q: 2 pmd: 1225722 q: 3 pmd: 648172 q: 281 pmd: 475560 q: 727 pmd: 983017
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An example first proved prime by Euler, illustrating the largest idoneal number. Thye output is quite wide, owing to the large numbers and five prime factors.
$$ 197^2 + 1848 \cdot 100^2 = 18518809 $$
n: 18518809 n-1: 18518808 = 2^3 3 7 11^2 911
2 3 7 11 911
a: 2 q: 2 pmd: 1 q: 3 pmd: 16231349 q: 7 pmd: 16379060 q: 11 pmd: 59331 q: 911 pmd: 3811171
a: 3 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 16379060 q: 11 pmd: 1593851 q: 911 pmd: 2997175
a: 4 q: 2 pmd: 1 q: 3 pmd: 2287459 q: 7 pmd: 9521077 q: 11 pmd: 1593851 q: 911 pmd: 258990
a: 5 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 11979317 q: 11 pmd: 1 q: 911 pmd: 3946179
a: 6 q: 2 pmd: 1 q: 3 pmd: 16231349 q: 7 pmd: 9521077 q: 11 pmd: 7734927 q: 911 pmd: 11712163
a: 7 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 6806466 q: 11 pmd: 1593851 q: 911 pmd: 3976121
a: 8 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 11979317 q: 11 pmd: 7734927 q: 911 pmd: 2657590
a: 9 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 9521077 q: 11 pmd: 6348008 q: 911 pmd: 9667332
a: 10 q: 2 pmd: 1 q: 3 pmd: 16231349 q: 7 pmd: 3830681 q: 11 pmd: 59331 q: 911 pmd: 12244102
a: 11 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 6806466 q: 11 pmd: 1113045 q: 911 pmd: 16378015
a: 12 q: 2 pmd: 1 q: 3 pmd: 2287459 q: 7 pmd: 11979317 q: 11 pmd: 6348008 q: 911 pmd: 3955206
a: 13 q: 2 pmd: 1 q: 3 pmd: 2287459 q: 7 pmd: 7039825 q: 11 pmd: 59331 q: 911 pmd: 2226290
a: 14 q: 2 pmd: 1 q: 3 pmd: 16231349 q: 7 pmd: 7039825 q: 11 pmd: 7734927 q: 911 pmd: 13425126
a: 15 q: 2 pmd: 1 q: 3 pmd: 1 q: 7 pmd: 3830681 q: 11 pmd: 1593851 q: 911 pmd: 18337913
a: 16 q: 2 pmd: 1 q: 3 pmd: 16231349 q: 7 pmd: 3830681 q: 11 pmd: 6348008 q: 911 pmd: 693902
a: 17 q: 2 pmd: 18518808 q: 3 pmd: 2287459 q: 7 pmd: 7039825 q: 11 pmd: 6348008 q: 911 pmd: 14940382
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https://en.wikipedia.org/wiki/Powerball
https://en.wikipedia.org/wiki/Modular_exponentiation
https://en.wikipedia.org/wiki/Lucas_primality_test
program C++ with GMP... Anything on a line after // is a comment, not affecting the program at all.
#include <iostream>
#include <stdlib.h>
#include <fstream>
#include <strstream>
#include <list>
#include <set>
#include <math.h>
#include <iomanip>
#include <string>
#include <algorithm>
#include <iterator>
#include <gmp.h>
#include <gmpxx.h>
#include "form.h"
using namespace std;
// g++ -o Lucas_Primality Lucas_Primality.cc -lgmp -lgmpxx
// set<mpz_class> div = mp_Divisors(
// set<mpz_class>::iterator iter;
// for(iter = div.begin(); iter != div.end(); ++iter)
// mpz_class mp_Raise_To_Power( mpz_class m, mpz_class n)
// set<mpz_class> mp_Prime_Divisors( mpz_class i)
// mp_powermod( mpz_class base, mpz_class exp, mpz_class mod)
// g++ -o Lucas_Primality Lucas_Primality.cc -lgmp -lgmpxx
int main()
{
cout << endl;
system("date");
cout << endl;
// ? n = 197^2 + 1848 * 100^2
// 18518809
mpz_class n = 18518809 ;
set<mpz_class> dive = mp_Prime_Divisors(n-1);
cout << n << " n-1: " << n-1 << " = " << mp_Factored(n-1) << endl;
set<mpz_class>::iterator iter;
for(iter = dive.begin(); iter != dive.end(); ++iter) cout << " "<< *iter ;
cout << " " << endl;
int goon = 1;
for( mpz_class a = 2; goon && a <= mp_Sqrt(n); ++a)
{
if( mp_coprime(a,n) )
{
int goon2 = 1;
cout << " a: " << a ;
for(iter = dive.begin(); iter != dive.end(); ++iter)
{
mpz_class q = *iter;
mpz_class w = mp_powermod(a, (n-1)/q, n);
cout << " q: " << q << " pmd: " << w ;
goon2 = goon2 & w != 1;
} // for iter
goon = !goon2;
cout << endl;
} // coprime
} // for a
cout << endl << endl;
// system("date");
return 0;
}
// g++ -o Lucas_Primality Lucas_Primality.cc -lgmp -lgmpxx
It does not really help you. Indeed, $5280=2^5\cdot 3\cdot 5\cdot 11$, so that you know that $2,3,5,11$ do not divide $5281$. However, up to the square root you have to test the primes up to $72$ to realize that $5281$ is a prime. In other words, you still have to test division by $$ 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71. $$
Edit: The $abc$-conjecture says something on the relationship between the prime decomposition of $a$, $b$ and $c=a+b$. If we have $a+b=c$ in integers, and $a,b$ contain high powers of primes, then $c$ should not do so. For example, $$ 2^{20}+3^{11}=1225723 $$ and $c=1225723$ contains only the first power of a prime. This is consistent with FLT, i.e., that $x^n+y^n$ should not be an $n$-th power $z^n$ for $n>2$.
As in Euclid's proof that there are infinitely many primes, you know that $p_1p_2\dots p_n+1$ is not divisible by any of $p_1,\dots,p_n$.
To check for primality you apparently still need to rule out all of the other possible prime factors, using the Sieve of Eratosthenes, say.
I would say that the answer to Gallian's question is yes. Though you still have a good bit of work left.