- #1
Amer
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is it possible to find a 6 Successive numbers like
x , x+1 , x+2 , x+3 ,x+4 ,x+5 such that one one is prime ?
Thanks
x , x+1 , x+2 , x+3 ,x+4 ,x+5 such that one one is prime ?
Thanks
Amer said:is it possible to find a 6 Successive numbers like
x , x+1 , x+2 , x+3 ,x+4 ,x+5 such that one one is prime ?
Thanks
chisigma said:Why don't try to generalize the problem: given k, how to compute an n such that n, n+1, n+2,...,n+k are all non prime numbers?...
chisigma said:An easy way to get the result in the particular case where k is prime is based on the consideration that $k|n \implies k|(n+m k)$. Setting $n = 2 \cdot 3 \cdot 5 \cdot ... \cdot k$ we are sure that $n+2,n+3,...,n+k+1$ are all non prime numbers. For example...
$\displaystyle k=11 \implies n=2310 \implies 2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322\ \text{are all non prime }$
Although 'easy' this method is often 'excessive' because the effective quantity consecutive non prime numbers can be greater. In the given example 2311 is prime so that the sequence starts at 2312 but 2323,2324,2325,2326,2327,2328,2329,2330,2331 and 2332 are non prime numbers [2333 is prime...] and the effective sequence's length is 20 [not 11]...
Kind regards
$\chi$ $\sigma$
Is it possible to find a 6 consecutive integers numbers like
x , x+1 , x+2 , x+3 ,x+4 ,x+5 such that not one is prime? . Yes!
soroban said:Hello, Amer!
One solution is: .[tex]x \:=\:7!+2[/tex]
. . [tex]\begin{array}{c}7!+2\text{ is divisible by 2} \\
7!+3\text{ is divisible by 3} \\ 7!+4\text{ is divisible by 4} \\
7!+5\text{ is divisible by 5} \\ 7!+6 \text{ is divisible by 6} \\
7!+7\text{ is divisible by 7} \\ \end{array}[/tex]There is a simpler (and much longer) list:
. . [tex]\begin{array}{ccc} 114 &=& 2\cdot57 \\ 115 &=& 5\cdot 23 \\ 116 &=& 2\cdot 58 \\ 117 &=& 3\cdot39 \\ 118 &=& 2\cdot59 \\ 119 &=& 7\cdot17 \\ 120 &=& 2\cdot60 \\ 121 &=& 11\cdot11 \\ 122 &=& 2\cdot61 \\ 123 &=& 3\cdot41 \\ 124 &=& 2\cdot62 \\ 125 &=& 5\cdot25 \\ 126 &=& 2\cdot63 \end{array}[/tex]
"6 Successive numbers no one is prime" refers to a sequence of six consecutive numbers where none of them are prime numbers. In other words, none of the numbers in the sequence can be divided evenly by any number other than 1 and itself.
Studying "6 Successive numbers no one is prime" can help us understand the patterns and properties of prime and composite numbers. It also allows us to identify the limitations and exceptions in mathematical concepts, leading to further exploration and advancements in the field.
A number is considered prime if it is only divisible by 1 and itself. On the other hand, a number is considered composite if it has more than two factors, meaning it can be divided evenly by numbers other than 1 and itself.
Yes, there are other sequences with similar properties, such as "7 Successive numbers no one is prime" and "8 Successive numbers no one is prime". These sequences can also be extended to any number of consecutive numbers.
No, this statement cannot be proven or disproven as there are infinite numbers and it is impossible to check every combination of six consecutive numbers. However, based on mathematical principles and patterns, it is highly likely that there are indeed "6 Successive numbers no one is prime".