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How-many-different-clock-type-dials-can-be-made-containing-first-n-natual-numbers-with-the-property-that-sum-of-any-two-numbers-of-consecutive-positions-be-a-prime-number-N-1-2-3-




Question Number 3172 by Rasheed Soomro last updated on 07/Dec/15
How many different  clock−type dials can be made   containing first n natual numbers with the property  that sum of  any two numbers of consecutive positions be  a prime number.  N={1,2,3,...}
$$\mathcal{H}{ow}\:{many}\:{different}\:\:{clock}−{type}\:{dials}\:{can}\:{be}\:{made}\: \\ $$$${containing}\:{first}\:{n}\:{natual}\:{numbers}\:{with}\:{the}\:{property} \\ $$$${that}\:{sum}\:{of}\:\:{any}\:{two}\:{numbers}\:{of}\:{consecutive}\:{positions}\:{be} \\ $$$${a}\:{prime}\:{number}. \\ $$$$\mathbb{N}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},…\right\} \\ $$
Commented by Rasheed Soomro last updated on 07/Dec/15
For n=1 there is no such dial.  For n=2 there is 1 such dial.    For n=3 there is no such dial.  For n=4 there are two such dials: 12341,14321   For n=5 :     ⋮  For n there are m such dials.   What is m?
$$\mathcal{F}{or}\:{n}=\mathrm{1}\:{there}\:{is}\:{no}\:{such}\:{dial}. \\ $$$$\mathcal{F}{or}\:{n}=\mathrm{2}\:{there}\:{is}\:\mathrm{1}\:{such}\:{dial}.\:\: \\ $$$$\mathcal{F}{or}\:{n}=\mathrm{3}\:{there}\:{is}\:{no}\:{such}\:{dial}. \\ $$$$\mathcal{F}{or}\:{n}=\mathrm{4}\:{there}\:{are}\:{two}\:{such}\:{dials}:\:\mathrm{12341},\mathrm{14321}\: \\ $$$$\mathcal{F}{or}\:{n}=\mathrm{5}\::\:\:\: \\ $$$$\vdots \\ $$$$\mathcal{F}{or}\:{n}\:{there}\:{are}\:{m}\:{such}\:{dials}.\:\:\:\mathcal{W}{hat}\:{is}\:{m}? \\ $$
Answered by prakash jain last updated on 06/Dec/15
No such formula is possible due to the fact that  there is no formula for prime numbers.
$$\mathrm{No}\:\mathrm{such}\:\mathrm{formula}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{due}\:\mathrm{to}\:\mathrm{the}\:\mathrm{fact}\:\mathrm{that} \\ $$$$\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{prime}\:\mathrm{numbers}. \\ $$
Commented by Rasheed Soomro last updated on 07/Dec/15
Any impossibiliy for some type of numbers? For   example for odd numbers etc?
$$\mathcal{A}{ny}\:{impossibiliy}\:{for}\:{some}\:{type}\:{of}\:{numbers}?\:{For}\: \\ $$$${example}\:{for}\:{odd}\:{numbers}\:{etc}? \\ $$

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