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How-many-solution-so-that-3n-4-4n-5-5n-13-are-prime-numbers-

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Question Number 73295 by naka3546 last updated on 10/Nov/19
How many solution so that 3n−4, 4n−5, 5n−13 are prime numbers ?
Howmanysolutionsothat3n−4,4n−5,5n−13areprimenumbers?
Answered by mind is power last updated on 10/Nov/19
(5n−13).(3n−4)=15n^2 −59n+52 =n^2 −n +14n^2 −58n+52=n(n−1)+2(7n^2 −29n+26)=2k ⇒2∣(5n−13)∨2∣(3n−4) 3n−4>2⇒n>2∣5n−13>2⇒n>3 if n>3 ⇒m=min (3n−4,5n−13)>2 since 3n−4 and 5n−13 are prim >2 ,⇒2 can not divide one of them ⇒n≤3 n=3⇒3n−4=5,4n−5=7,5n−13=2 all prime over N we have only one solution n=3 n≤2⇒5n−13<0 prim is defnd over N
(5n−13).(3n−4)=15n2−59n+52=n2−n+14n2−58n+52=n(n−1)+2(7n2−29n+26)=2k⇒2∣(5n−13)∨2∣(3n−4)3n−4>2⇒n>2∣5n−13>2⇒n>3ifn>3⇒m=min(3n−4,5n−13)>2since3n−4and5n−13areprim>2,⇒2cannotdivideoneofthem⇒n⩽3n=3⇒3n−4=5,4n−5=7,5n−13=2allprimeoverNwehaveonlyonesolutionn=3n⩽2⇒5n−13<0primisdefndoverN

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