How to finds primes $p$ with the property that both $10p^2+9$ and $8p^2-9$ are also primes using Wolfram...












1












$begingroup$


I want to know the number of primes $p$ that satisfies the condition that



$10p^2 + 9$



and



$8p^2 - 9$



are both primes, for primes $p$ only.



How can I do it using Wolfram Alpha (or any other online math software)?






polynomials prime-numbers wolfram-alpha







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 2:25










  • $begingroup$
    There probably are an infinite number of primes that satisfy this...
    $endgroup$
    – Don Thousand
    Aug 18 '18 at 2:30
















1












$begingroup$


I want to know the number of primes $p$ that satisfies the condition that



$10p^2 + 9$



and



$8p^2 - 9$



are both primes, for primes $p$ only.



How can I do it using Wolfram Alpha (or any other online math software)?






polynomials prime-numbers wolfram-alpha







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 2:25










  • $begingroup$
    There probably are an infinite number of primes that satisfy this...
    $endgroup$
    – Don Thousand
    Aug 18 '18 at 2:30














1












1








1





$begingroup$


I want to know the number of primes $p$ that satisfies the condition that



$10p^2 + 9$



and



$8p^2 - 9$



are both primes, for primes $p$ only.



How can I do it using Wolfram Alpha (or any other online math software)?






polynomials prime-numbers wolfram-alpha







share|cite|improve this question











$endgroup$




I want to know the number of primes $p$ that satisfies the condition that



$10p^2 + 9$



and



$8p^2 - 9$



are both primes, for primes $p$ only.



How can I do it using Wolfram Alpha (or any other online math software)?






polynomials prime-numbers wolfram-alpha




polynomials prime-numbers wolfram-alpha






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 18 '18 at 7:34









Asaf Karagila♦

308k33441774




308k33441774










asked Aug 18 '18 at 2:19









John FinkelsteinJohn Finkelstein

656




656








  • 1




    $begingroup$
    There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 2:25










  • $begingroup$
    There probably are an infinite number of primes that satisfy this...
    $endgroup$
    – Don Thousand
    Aug 18 '18 at 2:30














  • 1




    $begingroup$
    There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 2:25










  • $begingroup$
    There probably are an infinite number of primes that satisfy this...
    $endgroup$
    – Don Thousand
    Aug 18 '18 at 2:30








1




1




$begingroup$
There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
$endgroup$
– Toby Mak
Aug 18 '18 at 2:25




$begingroup$
There is a separate site for Mathematica, but if you want to ask about any other maths software (such as Maple), this is the right place.
$endgroup$
– Toby Mak
Aug 18 '18 at 2:25












$begingroup$
There probably are an infinite number of primes that satisfy this...
$endgroup$
– Don Thousand
Aug 18 '18 at 2:30




$begingroup$
There probably are an infinite number of primes that satisfy this...
$endgroup$
– Don Thousand
Aug 18 '18 at 2:30










3 Answers
3






active

oldest

votes


















5












$begingroup$

Using Wolfram Alpha/Mathematica, this is what I have so far:



10 * Prime[n]^2 + 9


calculates your expression for the $n$th prime.



PrimeQ[ 10 * Prime[n]^2 + 9 ]


checks if that expression is prime.



Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n,20} ]


makes a table of if that expression is prime or not from $1$ to $20$.



I tried:



Count [ Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n, 20} ], True ]


to count how many 'True's there were from $1$ to $20$.



This works on Wolfram Cloud.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Unquote True, it's a built-in symbol.
    $endgroup$
    – Robert Soupe
    Aug 18 '18 at 5:12










  • $begingroup$
    @RobertSoupe Wolfram Alpha still doesn't give an answer.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 9:17










  • $begingroup$
    Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
    $endgroup$
    – Robert Soupe
    Aug 19 '18 at 3:03



















2












$begingroup$

If you want to know how many such primes there are total, you're going to have to apply some good old fashioned human mathematical reasoning. Sometimes it's easy, sometimes it's hard.



e.g., how many primes consist of all 9s? That's easy, none in base 10. How many primes are of the form $2^p - 1$? That's hard. At least forty, but maybe not too many more than that.



But if you just want to know how many such of a given form there are in a reasonably small finite range, like, say, 1 to $10^{20}$, you can use Wolfram Alpha.



Many ways to skin a cat, Toby Mak has shown you a couple. Here's how I'd do it: first,



Select[10Prime[Range[100]]^2 + 9, PrimeQ]



Oops, I forgot you need to be a little more explicit in Wolfram Alpha:



Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]



First one should be 499, last one 2714419. Wrap that in Length like so:



Length[Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]]



In Wolfram Mathematica you can also do Length[%], which I read in my mind as "length of previous."



Answer is 21. Meaning that among the first hundred primes $p$, 21 of them are such that $10p^2 + 9$ is also prime.



You can push Wolfram Alpha a little higher than that, but not as high as you can push Wolfram Mathematica. Unless maybe you have a paid Wolfram Alpha subscription.



EDIT: Yong Hao Ng correctly points out that the asker wants $10p^2 + 9$ and $8p^2 - 9$. And also that that can be easily accommodated with an AND, e.g., Length[Select[Prime[Range[100]], PrimeQ[10#^2 + 9] && PrimeQ[8#^2 - 9] &]].






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
    $endgroup$
    – Yong Hao Ng
    Aug 18 '18 at 7:55



















0












$begingroup$

The answer to this question is no prime number can be formed satisfying your property because prime's can't be denoted in the subtraction of one- squared quantity and other cubed quantity.



At least, 10p^2 + 9 can never be satisfied.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    Could you provide more details?
    $endgroup$
    – klirk
    Dec 14 '18 at 12:00












Your Answer





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3 Answers
3






active

oldest

votes








3 Answers
3






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

Using Wolfram Alpha/Mathematica, this is what I have so far:



10 * Prime[n]^2 + 9


calculates your expression for the $n$th prime.



PrimeQ[ 10 * Prime[n]^2 + 9 ]


checks if that expression is prime.



Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n,20} ]


makes a table of if that expression is prime or not from $1$ to $20$.



I tried:



Count [ Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n, 20} ], True ]


to count how many 'True's there were from $1$ to $20$.



This works on Wolfram Cloud.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Unquote True, it's a built-in symbol.
    $endgroup$
    – Robert Soupe
    Aug 18 '18 at 5:12










  • $begingroup$
    @RobertSoupe Wolfram Alpha still doesn't give an answer.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 9:17










  • $begingroup$
    Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
    $endgroup$
    – Robert Soupe
    Aug 19 '18 at 3:03
















5












$begingroup$

Using Wolfram Alpha/Mathematica, this is what I have so far:



10 * Prime[n]^2 + 9


calculates your expression for the $n$th prime.



PrimeQ[ 10 * Prime[n]^2 + 9 ]


checks if that expression is prime.



Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n,20} ]


makes a table of if that expression is prime or not from $1$ to $20$.



I tried:



Count [ Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n, 20} ], True ]


to count how many 'True's there were from $1$ to $20$.



This works on Wolfram Cloud.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Unquote True, it's a built-in symbol.
    $endgroup$
    – Robert Soupe
    Aug 18 '18 at 5:12










  • $begingroup$
    @RobertSoupe Wolfram Alpha still doesn't give an answer.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 9:17










  • $begingroup$
    Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
    $endgroup$
    – Robert Soupe
    Aug 19 '18 at 3:03














5












5








5





$begingroup$

Using Wolfram Alpha/Mathematica, this is what I have so far:



10 * Prime[n]^2 + 9


calculates your expression for the $n$th prime.



PrimeQ[ 10 * Prime[n]^2 + 9 ]


checks if that expression is prime.



Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n,20} ]


makes a table of if that expression is prime or not from $1$ to $20$.



I tried:



Count [ Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n, 20} ], True ]


to count how many 'True's there were from $1$ to $20$.



This works on Wolfram Cloud.






share|cite|improve this answer











$endgroup$



Using Wolfram Alpha/Mathematica, this is what I have so far:



10 * Prime[n]^2 + 9


calculates your expression for the $n$th prime.



PrimeQ[ 10 * Prime[n]^2 + 9 ]


checks if that expression is prime.



Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n,20} ]


makes a table of if that expression is prime or not from $1$ to $20$.



I tried:



Count [ Table[ PrimeQ[ 10 * Prime[n]^2 + 9 ], {n, 20} ], True ]


to count how many 'True's there were from $1$ to $20$.



This works on Wolfram Cloud.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 19 '18 at 9:43

























answered Aug 18 '18 at 2:40









Toby MakToby Mak

3,57511128




3,57511128












  • $begingroup$
    Unquote True, it's a built-in symbol.
    $endgroup$
    – Robert Soupe
    Aug 18 '18 at 5:12










  • $begingroup$
    @RobertSoupe Wolfram Alpha still doesn't give an answer.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 9:17










  • $begingroup$
    Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
    $endgroup$
    – Robert Soupe
    Aug 19 '18 at 3:03


















  • $begingroup$
    Unquote True, it's a built-in symbol.
    $endgroup$
    – Robert Soupe
    Aug 18 '18 at 5:12










  • $begingroup$
    @RobertSoupe Wolfram Alpha still doesn't give an answer.
    $endgroup$
    – Toby Mak
    Aug 18 '18 at 9:17










  • $begingroup$
    Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
    $endgroup$
    – Robert Soupe
    Aug 19 '18 at 3:03
















$begingroup$
Unquote True, it's a built-in symbol.
$endgroup$
– Robert Soupe
Aug 18 '18 at 5:12




$begingroup$
Unquote True, it's a built-in symbol.
$endgroup$
– Robert Soupe
Aug 18 '18 at 5:12












$begingroup$
@RobertSoupe Wolfram Alpha still doesn't give an answer.
$endgroup$
– Toby Mak
Aug 18 '18 at 9:17




$begingroup$
@RobertSoupe Wolfram Alpha still doesn't give an answer.
$endgroup$
– Toby Mak
Aug 18 '18 at 9:17












$begingroup$
Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
$endgroup$
– Robert Soupe
Aug 19 '18 at 3:03




$begingroup$
Oh, yeah, sorry, I should have directed you to sandbox.open.wolframcloud.com instead.
$endgroup$
– Robert Soupe
Aug 19 '18 at 3:03











2












$begingroup$

If you want to know how many such primes there are total, you're going to have to apply some good old fashioned human mathematical reasoning. Sometimes it's easy, sometimes it's hard.



e.g., how many primes consist of all 9s? That's easy, none in base 10. How many primes are of the form $2^p - 1$? That's hard. At least forty, but maybe not too many more than that.



But if you just want to know how many such of a given form there are in a reasonably small finite range, like, say, 1 to $10^{20}$, you can use Wolfram Alpha.



Many ways to skin a cat, Toby Mak has shown you a couple. Here's how I'd do it: first,



Select[10Prime[Range[100]]^2 + 9, PrimeQ]



Oops, I forgot you need to be a little more explicit in Wolfram Alpha:



Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]



First one should be 499, last one 2714419. Wrap that in Length like so:



Length[Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]]



In Wolfram Mathematica you can also do Length[%], which I read in my mind as "length of previous."



Answer is 21. Meaning that among the first hundred primes $p$, 21 of them are such that $10p^2 + 9$ is also prime.



You can push Wolfram Alpha a little higher than that, but not as high as you can push Wolfram Mathematica. Unless maybe you have a paid Wolfram Alpha subscription.



EDIT: Yong Hao Ng correctly points out that the asker wants $10p^2 + 9$ and $8p^2 - 9$. And also that that can be easily accommodated with an AND, e.g., Length[Select[Prime[Range[100]], PrimeQ[10#^2 + 9] && PrimeQ[8#^2 - 9] &]].






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
    $endgroup$
    – Yong Hao Ng
    Aug 18 '18 at 7:55
















2












$begingroup$

If you want to know how many such primes there are total, you're going to have to apply some good old fashioned human mathematical reasoning. Sometimes it's easy, sometimes it's hard.



e.g., how many primes consist of all 9s? That's easy, none in base 10. How many primes are of the form $2^p - 1$? That's hard. At least forty, but maybe not too many more than that.



But if you just want to know how many such of a given form there are in a reasonably small finite range, like, say, 1 to $10^{20}$, you can use Wolfram Alpha.



Many ways to skin a cat, Toby Mak has shown you a couple. Here's how I'd do it: first,



Select[10Prime[Range[100]]^2 + 9, PrimeQ]



Oops, I forgot you need to be a little more explicit in Wolfram Alpha:



Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]



First one should be 499, last one 2714419. Wrap that in Length like so:



Length[Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]]



In Wolfram Mathematica you can also do Length[%], which I read in my mind as "length of previous."



Answer is 21. Meaning that among the first hundred primes $p$, 21 of them are such that $10p^2 + 9$ is also prime.



You can push Wolfram Alpha a little higher than that, but not as high as you can push Wolfram Mathematica. Unless maybe you have a paid Wolfram Alpha subscription.



EDIT: Yong Hao Ng correctly points out that the asker wants $10p^2 + 9$ and $8p^2 - 9$. And also that that can be easily accommodated with an AND, e.g., Length[Select[Prime[Range[100]], PrimeQ[10#^2 + 9] && PrimeQ[8#^2 - 9] &]].






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
    $endgroup$
    – Yong Hao Ng
    Aug 18 '18 at 7:55














2












2








2





$begingroup$

If you want to know how many such primes there are total, you're going to have to apply some good old fashioned human mathematical reasoning. Sometimes it's easy, sometimes it's hard.



e.g., how many primes consist of all 9s? That's easy, none in base 10. How many primes are of the form $2^p - 1$? That's hard. At least forty, but maybe not too many more than that.



But if you just want to know how many such of a given form there are in a reasonably small finite range, like, say, 1 to $10^{20}$, you can use Wolfram Alpha.



Many ways to skin a cat, Toby Mak has shown you a couple. Here's how I'd do it: first,



Select[10Prime[Range[100]]^2 + 9, PrimeQ]



Oops, I forgot you need to be a little more explicit in Wolfram Alpha:



Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]



First one should be 499, last one 2714419. Wrap that in Length like so:



Length[Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]]



In Wolfram Mathematica you can also do Length[%], which I read in my mind as "length of previous."



Answer is 21. Meaning that among the first hundred primes $p$, 21 of them are such that $10p^2 + 9$ is also prime.



You can push Wolfram Alpha a little higher than that, but not as high as you can push Wolfram Mathematica. Unless maybe you have a paid Wolfram Alpha subscription.



EDIT: Yong Hao Ng correctly points out that the asker wants $10p^2 + 9$ and $8p^2 - 9$. And also that that can be easily accommodated with an AND, e.g., Length[Select[Prime[Range[100]], PrimeQ[10#^2 + 9] && PrimeQ[8#^2 - 9] &]].






share|cite|improve this answer











$endgroup$



If you want to know how many such primes there are total, you're going to have to apply some good old fashioned human mathematical reasoning. Sometimes it's easy, sometimes it's hard.



e.g., how many primes consist of all 9s? That's easy, none in base 10. How many primes are of the form $2^p - 1$? That's hard. At least forty, but maybe not too many more than that.



But if you just want to know how many such of a given form there are in a reasonably small finite range, like, say, 1 to $10^{20}$, you can use Wolfram Alpha.



Many ways to skin a cat, Toby Mak has shown you a couple. Here's how I'd do it: first,



Select[10Prime[Range[100]]^2 + 9, PrimeQ]



Oops, I forgot you need to be a little more explicit in Wolfram Alpha:



Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]



First one should be 499, last one 2714419. Wrap that in Length like so:



Length[Select[10Prime[Range[100]]^2 + 9, PrimeQ[#] &]]



In Wolfram Mathematica you can also do Length[%], which I read in my mind as "length of previous."



Answer is 21. Meaning that among the first hundred primes $p$, 21 of them are such that $10p^2 + 9$ is also prime.



You can push Wolfram Alpha a little higher than that, but not as high as you can push Wolfram Mathematica. Unless maybe you have a paid Wolfram Alpha subscription.



EDIT: Yong Hao Ng correctly points out that the asker wants $10p^2 + 9$ and $8p^2 - 9$. And also that that can be easily accommodated with an AND, e.g., Length[Select[Prime[Range[100]], PrimeQ[10#^2 + 9] && PrimeQ[8#^2 - 9] &]].







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 18 '18 at 8:37

























answered Aug 18 '18 at 5:33









Robert SoupeRobert Soupe

11.5k21950




11.5k21950








  • 1




    $begingroup$
    A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
    $endgroup$
    – Yong Hao Ng
    Aug 18 '18 at 7:55














  • 1




    $begingroup$
    A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
    $endgroup$
    – Yong Hao Ng
    Aug 18 '18 at 7:55








1




1




$begingroup$
A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
$endgroup$
– Yong Hao Ng
Aug 18 '18 at 7:55




$begingroup$
A remark that for both conditions one could do Length[Select[Prime[Range[100]], PrimeQ[10#^2+9]&&PrimeQ[8#^2-9]&]]
$endgroup$
– Yong Hao Ng
Aug 18 '18 at 7:55











0












$begingroup$

The answer to this question is no prime number can be formed satisfying your property because prime's can't be denoted in the subtraction of one- squared quantity and other cubed quantity.



At least, 10p^2 + 9 can never be satisfied.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    Could you provide more details?
    $endgroup$
    – klirk
    Dec 14 '18 at 12:00
















0












$begingroup$

The answer to this question is no prime number can be formed satisfying your property because prime's can't be denoted in the subtraction of one- squared quantity and other cubed quantity.



At least, 10p^2 + 9 can never be satisfied.






share|cite|improve this answer









$endgroup$









  • 2




    $begingroup$
    Could you provide more details?
    $endgroup$
    – klirk
    Dec 14 '18 at 12:00














0












0








0





$begingroup$

The answer to this question is no prime number can be formed satisfying your property because prime's can't be denoted in the subtraction of one- squared quantity and other cubed quantity.



At least, 10p^2 + 9 can never be satisfied.






share|cite|improve this answer









$endgroup$



The answer to this question is no prime number can be formed satisfying your property because prime's can't be denoted in the subtraction of one- squared quantity and other cubed quantity.



At least, 10p^2 + 9 can never be satisfied.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 14 '18 at 10:11









Aziz LokhandwalaAziz Lokhandwala

14




14








  • 2




    $begingroup$
    Could you provide more details?
    $endgroup$
    – klirk
    Dec 14 '18 at 12:00














  • 2




    $begingroup$
    Could you provide more details?
    $endgroup$
    – klirk
    Dec 14 '18 at 12:00








2




2




$begingroup$
Could you provide more details?
$endgroup$
– klirk
Dec 14 '18 at 12:00




$begingroup$
Could you provide more details?
$endgroup$
– klirk
Dec 14 '18 at 12:00


















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