# Do we share the prime cluster?

The prime cluster of an integer N higher than 2 is defined as the pair formed by the highest prime strictly lower than N and the lowest prime strictly higher than N.

Note that following the definition above, if the integer is a prime itself, then its prime cluster is the pair of the primes preceding and succeeding it.

Given two integers integers N, M (N, M ≥ 3), output a truthy / falsy value based on whether N and M have the same prime cluster.

This is , so the aim is to reduce your byte count as much as possible. Thus, the shortest code in every programming language wins.

# Test cases / Examples

For instance, the prime cluster of 9 is [7, 11], because:

• 7 is the highest prime strictly lower than 9, and
• 11 is the lowest prime strictly higher than 9.

Similarly, the the prime cluster of 67 is [61, 71] (note that 67 is a prime).

Truthy pairs

8, 10
20, 22
65, 65
73, 73
86, 84
326, 318
513, 518


Falsy pairs

4, 5
6, 8
409, 401
348, 347
419, 418
311, 313
326, 305

• Do the truthy / falsy values have to be two distinct values or can one define a mapping from their program's output to a truthy / falsy value and output (potentially infinitely) many different values? – Jonathan Frech Nov 7 '17 at 22:53
• @JonathanFrech Truthy/Falsy per decision-problem definition, not necessarily consistent but distict and truthy/falsy – Mr. Xcoder Nov 8 '17 at 5:03

# Jelly, 6435 4 bytes

rÆPE


### How it works

rÆPE    Main link. Arguments: N, M
r       Yield the range of integers between N and M, inclusive.
ÆP     For each integer, yield 1 if it is prime, 0 otherwise.
E    Yield 1 if all items are equal (none in the range were prime,
or there's only one item).


Works because two numbers have different prime clusters iff there is a prime between them, or either number is itself prime; unless both numbers are the same, in which case E returns 1 anyway (all items in a single-item array are equal).

• Your programs source doesn’t look friendly... – Stan Strum Nov 7 '17 at 21:05

# Perl 6, 52 bytes

{[eqv] @_».&{(($_...0),$_..*)».first(*.is-prime)}}


Test it

## Expanded:

{  # bare block lambda with implicit slurpy input ｢@_｣

[eqv]               # see if each sub list is equivalent

@_».&{            # for each value in the input

(

( $_ ... 0 ), # decreasing Seq$_ ..  *    # Range

)».first(*.is-prime) # find the first prime from both the Seq and Range

}
}


# Python 3, 10395 91 bytes

lambda*z:len({*z})<2or[1for i in range(min(z),max(z)+1)if all(i%k for k in range(2,i))]<[0]


Try it online!

->n,m{[*n..m,*m..n].all?{|x|?1*x=~/^(11+)\1+$/}||n==m}  Try it online! Uses the horrible regex primality test from my answer (which I had forgotten about until I clicked on it) to the related question Is this number a prime?. Since we have N, M ≥ 3, the check for 1 can be removed from the pattern, making the byte count less than using the built-in. Note: The regex primality test is pathologically, hilariously inefficient. I believe it's at least O(n!), though I don't have time to figure it right now. It took twelve seconds for it to check 100,001, and was grinding for five or ten minutes on 1,000,001 before I canceled it. Use/abuse at your own risk. • At that rate it is likely n². You know, 100001! = 2824257650254427477772164512240315763832679701040485762827423875723843380680572028502730496931545301922349718873479336571104510933085749261906300669827923360329777024436472705878118321875571799283167659071802605510878659379955675120386166847407407122463765792082065493877636247683663198828626954833262077780844919163487776145463353109634071852657157707925315037717734498612061347682956332369235999129371094504360348686870713719732258380465223614176068 ... (Warning: The output exceeded 128 KiB and was truncated.) which will take millenia to run. – user202729 Nov 7 '17 at 1:10 # Retina, 58 bytes \b(.+)¶\1\b .+$*
O
+\b(1+)¶11\1
$1¶1$&
A^(11+)\1+$^$


Try it online! Explanation:

\b(.+)¶\1\b


If both inputs are the same, simply delete everything, and fall through to output 1 at the end.

.+
$*  Convert to unary. O  Sort into order. +\b(1+)¶11\1$1¶1$&  Expand to a range of all the numbers. A^(11+)\1+$


Delete all composite numbers.

^\$


If there are no numbers left, output 1, otherwise 0.

# PARI/GP, 28 bytes

v->s=Set(v);#s<2||!primes(s)


Try it online with all test cases!

Returns 0 or 1 (usual PARI/GP "Boolean" values).

Explanation:

v must be a vector (or a column vector, or a list) with the two numbers N and M as coordinates. For example [8, 10]. Then s will be the "set" made from these numbers, which is either a one-coordinate vector (if N==M), or a two-coordinate vector with sorted entries otherwise.

Then if the number #s of coordinates in s is just one, we get 1 (truthy). Otherwise, primes will return a vector of all primes in the closed interval from s[1] to s[2]. Negation ! of that will give 1 if the vector is empty, while negation of a vector of one or more non-zero entries (here one or more primes) will give 0.

# JavaScript (ES6), 57 56 bytes

Takes input in currying syntax (a)(b). Returns 0 or 1.

a=>b=>a==b|!(g=k=>a%--k?g(k):k<2||a-b&&g(a+=a<b||-1))(a)


### Test cases

let f =

a=>b=>a==b|!(g=k=>a%--k?g(k):k<2||a-b&&g(a+=a<b||-1))(a)

console.log('Truthy')
console.log(f(8)(10))
console.log(f(20)(22))
console.log(f(65)(65))
console.log(f(73)(73))
console.log(f(86)(84))
console.log(f(326)(318))
console.log(f(513)(518))

console.log('Falsy')
console.log(f(4)(5))
console.log(f(6)(8))
console.log(f(409)(401))
console.log(f(348)(347))
console.log(f(419)(418))
console.log(f(311)(313))

### How?

a => b =>                 // given a and b
a == b |                // if a equals b, force success right away
!(g = k =>              // g = recursive function taking k
a % --k ?             //   decrement k; if k doesn't divide a:
g(k)                //     recursive calls until it does
:                     //   else:
k < 2 ||            //     if k = 1: a is prime -> return true (failure)
a - b &&            //     if a equals b: neither the original input integers nor
//     any integer between them are prime -> return 0 (success)
g(a += a < b || -1) //     else: recursive call with a moving towards b
)(a)                    // initial call to g()


# R, 63 46 bytes

-17 by Giuseppe

function(a,b)!sd(range(numbers::isPrime(a:b)))


Try it online!

Pretty simple application of ETHProductions' Jelly solution. Main interesting takeaway is was that with R boolean vectors any(x)==all(x) is equivalent to min(x)==max(x).

# C (gcc), 153 146 bytes

i,B;n(j){for(B=i=2;i<j;)B*=j%i++>0;return!B;}
#define g(l,m,o)for(l=o;n(--l););for(m=o;n(++m););
a;b;c;d;h(e,f){g(a,b,e)g(c,d,f)return!(a-c|b-d);}


-7 from Jonathan Frech

Defines a function h which takes in two ints and returns 1 for truthy and 0 for falsey

Try it online!

n is a function that returns 1 if its argument is not prime.

g is a macro that sets its first and second arguments to the next prime less than and greater than (respectively) it's third argument

h does g for both inputs and checks whether the outputs are the same.

• return a==c&&b==d; can be return!(a-c|b-d);. – Jonathan Frech Nov 6 '17 at 21:46
• – Jonathan Frech Nov 6 '17 at 21:53
• @JonathanFrech Fixed the TIO link. – pizzapants184 Nov 7 '17 at 22:15

# Jelly, 6 bytes

ÆpżÆnE


Try it online!

-2 thanks to Dennis.

# APL (Dyalog Unicode), 18+16 = 34 24 bytes

⎕CY'dfns'
∧/=/4 ¯4∘.pco⎕


Try it online!

Thanks to Adám for 10 bytes.

The line ⎕CY'dfns' (COPY) is needed to import the dfns (dynamic functions) collection, included with default Dyalog APL installs.

### How it works:

∧/=/4 ¯4∘.pco⎕ ⍝ Main function. This is a tradfn body.
⎕ ⍝ The 'quad' takes the input (in this case, 2 integers separated by a comma.
pco  ⍝ The 'p-colon' function, based on p: in J. Used to work with primes.
4 ¯4∘.     ⍝ Applies 4pco (first prime greater than) and ¯4pco (first prime smaller than) to each argument.
=/           ⍝ Compares the two items on each row
∧/             ⍝ Applies the logical AND between the results.
⍝ This yields 1 iff the prime clusters are equal.


# Python 2, 87 86 bytes

lambda*v:v[0]==v[1]or{1}-{all(v%i for i in range(2,v))for v in range(min(v),max(v)+1)}


Try it online!

• I like your set usage, even though it is not required for 87 bytes. – Jonathan Frech Nov 7 '17 at 21:19
• @JonathanFrech I got it to 86 using sets – ovs Nov 7 '17 at 21:41

# C (gcc), 103 bytes 100 bytes

i,j,p,s;f(m,n){s=1;for(i=m>n?i=n,n=m,m=i:m;i<=n;i++,p?s=m==n:0)for(p=j=2;j<i;)p=i%j++?p:0;return s;}


Try it online!

A straightforward solution:

p z=[x|x<-z,all((0/=).mod x)[2..x-1]]!!0
c x=(p[x-1,x-2..],p[x+1..])
x!y=c x==c y


Try it online!

# Mathematica, 3927 26 bytes

Equal@@#~NextPrime~{-1,1}&


Expanded:

                         &  # pure function, takes 2-member list as input
#~NextPrime~{-1,1}   # infix version of NextPrime[#,{-1,1}], which
# finds the upper and lower bounds of each
argument's prime clusters
Equal@@                     # are those bounds pairs equal?


Usage:

Equal@@#~NextPrime~{-1,1}& [{8, 10}]
(*  True  *)

Equal@@#~NextPrime~{-1,1}& [{6, 8}]
(*  False  *)

Equal@@#~NextPrime~{-1,1}& /@ {{8, 10}, {20, 22}, {65, 65},
{73, 73}, {86, 84}, {326, 318}, {513, 518}}
(*  {True, True, True, True, True, True, True}  *)

Equal@@#~NextPrime~{-1,1}& /@ {{4, 5}, {6, 8}, {409, 401},
{348, 347}, {419, 418}, {311, 313}}
(*  {False, False, False, False, False, False}  *)


Contributions: -12 bytes by Jenny_mathy, -1 byte by Martin Ender

• This only checks next prime. Try NextPrime[#,{-1,1}] – J42161217 Nov 6 '17 at 20:10
• @Jenny_mathy : I see you are correct. Caught by the "348, 347" test case, which is now demonstrated to pass. – Eric Towers Nov 6 '17 at 23:28
• 27 bytes: Equal@@NextPrime[#,{-1,1}]& takes as input [{N,M}] or if you want to keep the original input use this 30 bytes: Equal@@NextPrime[{##},{-1,1}]& – J42161217 Nov 7 '17 at 1:18
• @Jenny_mathy : Well, ..., the specified input is two integers, not a list, so ... – Eric Towers Nov 7 '17 at 2:43
• @EricTowers taking a list is fine. Also, you can save a byte by using infix notation #~NextPrime~{-1,1}. – Martin Ender Nov 7 '17 at 9:09

# J, 15 bytes

-:&(_4&p:,4&p:)


How it works:

   &(           ) - applies the verb in the brackets to both arguments
4&p:  - The smallest prime larger than y
_4&p:       - The largest prime smaller than y
,      - append
-:               - matches the pairs of the primes


Try it online!