Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$
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Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number is small enough for ECM, but this is quite slow for numbers
of this magnitude.
Is there anything better than trial division or ECM ?
number-theory prime-factorization tetration
edited Dec 11 '18 at 14:55
Klangen
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
$endgroup$
add a comment |
$begingroup$
Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number is small enough for ECM, but this is quite slow for numbers
of this magnitude.
Is there anything better than trial division or ECM ?
number-theory prime-factorization tetration
edited Dec 11 '18 at 14:55
Klangen
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
$endgroup$
2
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
1
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
$endgroup$
– apnorton
Feb 20 '14 at 1:16
add a comment |
$begingroup$
Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number is small enough for ECM, but this is quite slow for numbers
of this magnitude.
Is there anything better than trial division or ECM ?
number-theory prime-factorization tetration
edited Dec 11 '18 at 14:55
Klangen
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
$endgroup$
Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number is small enough for ECM, but this is quite slow for numbers
of this magnitude.
Is there anything better than trial division or ECM ?
number-theory prime-factorization tetration
number-theory prime-factorization tetration
edited Dec 11 '18 at 14:55
Klangen
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
edited Dec 11 '18 at 14:55
Klangen
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
edited Dec 11 '18 at 14:55
Klangen
1,72811334
edited Dec 11 '18 at 14:55
Klangen
1,72811334
edited Dec 11 '18 at 14:55
Klangen
1,72811334
1,72811334
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
asked Feb 19 '14 at 23:08
PeterPeter
47.4k1039129
47.4k1039129
2
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
1
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
$endgroup$
– apnorton
Feb 20 '14 at 1:16
add a comment |
2
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
1
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
$endgroup$
– apnorton
Feb 20 '14 at 1:16
2
2
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
1
1
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)$endgroup$
– apnorton
Feb 20 '14 at 1:16
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)$endgroup$
– apnorton
Feb 20 '14 at 1:16
add a comment |
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2
$begingroup$
'The number is small enough for ECM' - this number is roughly 700K decimal digits; AFAIK that's still well out of reach of even the best general-purpose factoring methods...
$endgroup$
– Steven Stadnicki
Feb 19 '14 at 23:17
1
$begingroup$
For people like me who like W|A... it choked on this.
$endgroup$
– apnorton
Feb 19 '14 at 23:35
$begingroup$
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)$endgroup$
– apnorton
Feb 20 '14 at 1:16