what is the biggest prime p verifying: p=Σ_(k=1) ^n [(√k)] where n∈N and [x] is floor(x)


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Question Number 88207 by MAB last updated on 09/Apr/20

$w h a t i s t h e b i g g e s t p r i m e p v e r i f y i n g :$ $p = \underset{k = 1}{\sum^{n}} [\sqrt{k}]$ $w h e r e n \in N and [x] i s f l o o r (x)$
Commented by MJS last updated on 09/Apr/20

$seems to be 197$ $I^{'} m working on an explanation$
	Answered by MJS last updated on 09/Apr/20

	$r = [\sqrt{k}]; k, q \in N$ $1 ⩽ k < 4 \Rightarrow r = 1$ $4 ⩽ k < 9 \Rightarrow r = 2$ $9 ⩽ k < 16 \Rightarrow r = 3$ $q^{2} ⩽ k < {(q + 1)}^{2} \Rightarrow r = q$ $we get q [2 q + 1] times$ $look at the sums p_{q} = \underset{k = 1}{\sum^{{(q + 1)}^{2} - 1}} [\sqrt{k}]$ $p_{1} = 3$ $now we start adding 2 s \Rightarrow uneven p^{'} s$ $p_{2} = 13$ $now adding 3 s \Rightarrow even & uneven p^{'} s$ $p_{3} = 34 = 2 \times 17$ $now adding 4 s \Rightarrow even p^{'} s$ $p_{4} = 70 = 2 \times 5 \times 7$ $now adding 5 s \Rightarrow 5 ∣ p$ $p_{5} = 125 = 5^{3}$ $now adding 6 s \Rightarrow even & uneven p^{'} s$ $p_{6} = 203 = 7 \times 29$ $now adding 7 s \Rightarrow 7 ∣ p$ $p_{7} = 308 = 2^{2} \times 7 \times 11$ $now adding 8 s \Rightarrow even p^{'} s$ $p_{8} = 444 = 2^{2} \times 3 \times 37$ $now adding 9 s \Rightarrow 3 ∣ p$ $p_{9} = 615 = 3 \times 5 \times 41$ $now adding 10 s \Rightarrow 5 ∣ p$ $p_{10} = 825 = 3 \times 5^{2} \times 11$ $now adding 11 s \Rightarrow 11 ∣ p$ $. . .$ $I {don}^{'} t have the time to continue$ $we must prove that \gcd (p_{q}, q + 1) \neq 1 \forall q > 6$
	Commented by MAB last updated on 18/May/20

	$y o u a r e i n t h e r i g h t w a y, c o n t i n u e .$ $h i n t : w h y {d o n}^{'} t y o u u s e v a l u a t i o n ?$
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