Description

Where ? and ? are constants (0 < ? < ?) to show that there exists a constant ? > 1 such that for every ? ≥ 2 there is a prime satisfying ? < ? ≤ ??.

Solution. Let ? be any number such that ? > ?/?. Then ∑ log ? ?<?≤?? = ∑ log ? ?≤?? − ∑ log ? ?≤? ≥ ?(??)− ?? > ?? − ?? = 0 Since this sum is non-zero, there exists a prime ? satisfying ? < ? ≤ ??. Let us show now that ? = ?/? also satisfies the condition. Fix any ? ≥ 2. As it is proved, the interval (?, (1 + ?/?)?] contains at least one prime number. Let ? be the minimal prime in this interval. Suppose ? > (?/?)?. Then ? = 1 2 ( ? ? + ? ? ) > 1 2 ( ? ? + ? ? ) = ? ? and ? = 1 2 ( ? ? + ? ? ) < 1 2 ( ? ? + ? ? ) = ? � 

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