Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$
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
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
add a comment |
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
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
2
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
1
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
– apnorton
Feb 20 '14 at 1:16
add a comment |
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
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
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
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
edited Dec 11 '18 at 14:55
Klangen
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
edited Dec 11 '18 at 14:55
Klangen
1,65711334
edited Dec 11 '18 at 14:55
Klangen
1,65711334
edited Dec 11 '18 at 14:55
Klangen
1,65711334
1,65711334
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
asked Feb 19 '14 at 23:08
Peter
46.7k1039125
46.7k1039125
2
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
1
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
– apnorton
Feb 20 '14 at 1:16
add a comment |
2
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
1
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.:)
– apnorton
Feb 20 '14 at 1:16
2
2
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
1
1
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)– apnorton
Feb 20 '14 at 1:16
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)– apnorton
Feb 20 '14 at 1:16
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f682779%2fprime-factors-of-222333555777%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f682779%2fprime-factors-of-222333555777%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
'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...
– Steven Stadnicki
Feb 19 '14 at 23:17
1
For people like me who like W|A... it choked on this.
– apnorton
Feb 19 '14 at 23:35
Interestingly enough, the sum of the first two addends is prime. Yet I entirely fail to see how that helps you in any way.
:)– apnorton
Feb 20 '14 at 1:16