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#### checkittwice

##### Member

- Apr 3, 2012

- 37

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[tex]Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.[/tex]

Examples:

[tex]p_1 \ = \ 2[/tex]

[tex]p_2 \ = \ 3[/tex]

[tex]p_3 \ = \ 5[/tex]

[tex]p_4 \ = \ 7[/tex]

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[tex]Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers.[/tex]

[tex]p_n \ + \ p_{n + 1} \ \ \ge \ \ p_{n + 2} \ + \ p_{n - 2}, \ \ \ for \ \ all \ \ n \ \ge \ 4.[/tex]

Examples:

[tex] \ \ 7 \ + \ 11 \ \ > \ \ 13 \ + \ \ 3[/tex]

[tex]19 \ + \ 23 \ \ = \ \ 29 \ + \ 13[/tex]

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[tex]Let \ \ p_n \ \ = \ \ the \ \ nth \ \ prime \ \ number.[/tex]

Examples:

[tex]p_1 \ = \ 2[/tex]

[tex]p_2 \ = \ 3[/tex]

[tex]p_3 \ = \ 5[/tex]

[tex]p_4 \ = \ 7[/tex]

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[tex]Let \ \ n \ \ belong \ \ to \ \ the \ \ set \ \ of \ \ positive \ \ integers.[/tex]

**Prove (or disprove) the following:**[tex]p_n \ + \ p_{n + 1} \ \ \ge \ \ p_{n + 2} \ + \ p_{n - 2}, \ \ \ for \ \ all \ \ n \ \ge \ 4.[/tex]

Examples:

[tex] \ \ 7 \ + \ 11 \ \ > \ \ 13 \ + \ \ 3[/tex]

[tex]19 \ + \ 23 \ \ = \ \ 29 \ + \ 13[/tex]

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