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צפה בגרסה המלאה : [אלגברה] מספרים ראשוניים



אריאל
02-10-2009, 12:02
נתון n הוא מספר טבעי.

עבור אילו nים הביטוי n^4+4 הוא מספר ראשוני?

ניחושים לא מתקבלים ( שכן זו לא הוכחה.. )

בהצלחה !

Hurricane
02-10-2009, 14:20
קודם כל, n חייב להיות אי זוגי.
בדקתי עם מחשב עד n=1000 ומצאתי שזה מתקיים רק כאשר n=1.

עכשיו רק צריך למצוא הוכחה. :P

-רון-
02-10-2009, 14:54
לא מכל מספר ראשוני שמפחיתים ממנו 4 מקבלים מספר ראשוני נוסף! :)

lori
02-10-2009, 14:56
אה באמת? אפשר דוגמה? :)

Hurricane
02-10-2009, 14:58
5 ו- 13.


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9133 is prime.
9137 is prime.
9151 is prime.
9157 is prime.
9161 is prime.
9173 is prime.
9181 is prime.
9187 is prime.
9199 is prime.
9203 is prime.
9209 is prime.
9221 is prime.
9227 is prime.
9239 is prime.
9241 is prime.
9257 is prime.
9277 is prime.
9281 is prime.
9283 is prime.
9293 is prime.
9311 is prime.
9319 is prime.
9323 is prime.
9337 is prime.
9341 is prime.
9343 is prime.
9349 is prime.
9371 is prime.
9377 is prime.
9391 is prime.
9397 is prime.
9403 is prime.
9413 is prime.
9419 is prime.
9421 is prime.
9431 is prime.
9433 is prime.
9437 is prime.
9439 is prime.
9461 is prime.
9463 is prime.
9467 is prime.
9473 is prime.
9479 is prime.
9491 is prime.
9497 is prime.
9511 is prime.
9521 is prime.
9533 is prime.
9539 is prime.
9547 is prime.
9551 is prime.
9587 is prime.
9601 is prime.
9613 is prime.
9619 is prime.
9623 is prime.
9629 is prime.
9631 is prime.
9643 is prime.
9649 is prime.
9661 is prime.
9677 is prime.
9679 is prime.
9689 is prime.
9697 is prime.
9719 is prime.
9721 is prime.
9733 is prime.
9739 is prime.
9743 is prime.
9749 is prime.
9767 is prime.
9769 is prime.
9781 is prime.
9787 is prime.
9791 is prime.
9803 is prime.
9811 is prime.
9817 is prime.
9829 is prime.
9833 is prime.
9839 is prime.
9851 is prime.
9857 is prime.
9859 is prime.
9871 is prime.
9883 is prime.
9887 is prime.
9901 is prime.
9907 is prime.
9923 is prime.
9929 is prime.
9931 is prime.
9941 is prime.
9949 is prime.
9967 is prime.
9973 is prime.

אריאל
02-10-2009, 15:01
לורי ניסיון נחמד, אך כפי שרון אמר, אחת הטענות שלך שגויות.

נסו לפרק את הגורם n^4+4 לשני גורמים ולראות מה קורה בעקבות זה..

בהצלחה

lori
02-10-2009, 15:18
האם למספר לא ראשוני שמחברים 4 יוצא מספר לא ראשוני כמעט בכל המקרים?

אריאל
02-10-2009, 15:21
כן ו..?
ומה זה "כמעט בכל המקרים" יש אינסוף מקרים שזה קורה, ויש אינסוף מקרים שזה לא קורה..
בכל מקרה אני לא רואה איך זה קשור לכאן

גל_כהן
02-10-2009, 15:46
על פי הזהות של סופי ז'רמן (Sophie Germain Identity) מתקיים :
x^4+4y^4=(x^2+2y^2+2xy)(x^2+2y^2-2xy) ובמקרה הנ"ל :
x=n \ , \ y=1 ולכן :
n^4+4=(n^2+2+2n)(n^2+2-2n) ולאחר סידור :
n^4+4=(n^2+2n+2)(n^2-2n+2).
כעת קל לראות שרק במקרה בו אחד הגורמים הוא ראשוני והגורם השני הוא 1,
המספר כולו הוא ראשוני.
נבדוק מתי הגורם הראשון הוא 1 ומתי הגורם השני הוא 1.
n^2+2n+2=1 \right n=-1, אבל n הוא טבעי, ולכן פתרון זה נפסל.
n^2-2n+2=1 \right n=1 ועבור אפשרות זו מתקבל n^4+4=5,
שהוא כמובן ראשוני.
עבור כל אופציה אחרת מקבלים מכפלה של שני מספרים (ראשוניים או לאו),
ומספר שמתפרק למכפלה של לפחות שני מספרים הוא פריק.

יום טוב :) !

אריאל
02-10-2009, 15:55
פתרון מדויק גל !

אגב לפירוק שרשמת ניתן להגיע על ידי :

n^4+4=n^4+4n^2+4-4n^2=(n^2+2)^2-(2n)^2=[(n^2+2)+2n)][(n^2+2)-2n]=(n^2+2n+2)(n^2-2n+2)

גל_כהן
02-10-2009, 16:43
אריאל, זה בדיוק הרעיון בזהות של סופי ז'רמן...

יום טוב :) !