Problems & Puzzles: Puzzles

Puzzle 255.  k*p+1

Norman Luhn has published that:

2481129210 x p - 1 is prime for p = 2, 3, 5, 7, 11, 13, 17, 19, 23 or 29 (the first 10 primes!)

But I have not found the counterpart to this curio, that is to say what is the minimal k value such that k.p +1 is prime for the first 10 primes p.

Q. Can you find such k earliest value?

Solution: Contributions came form Patrick De Geest, Adam Stinchcombe, Jon Wharf, Faride Firoozbakht, J. C. Rosa and Ken Wilke.

Patrick and Faride just found that the asked question was solved by the sequence A084700 reported by Amarnat Murphy, and then k= 37202202450

Stinchcombe, Rosa and Wilke devised a smart approach in order to calculate not by an exhaustive search the asked question:

Stinchcombe wrote:

I find k=37202202450 as the first value.  I first sieved and found 25200 values of k mod m=2*3*5*7*11*13*17*19*23 which were good candidates for k (i.e., had k*p+1 divisible by neither 2,3,5,7,11,13,17,19, nor 23), then searched the residues classes for actual values of k.  I did not include the factor 29 in the modulus because of concern over memory requirements.

Rosa wrote

I have found that the minimal k value such that k.p+1 is prime for the first 10 primes is: k=37202202450=2.3.3.3.3.5.5.7.17.77191... I have found the minimal k value such that k.p+1 is prime for the first 11 primes . It is : k=148684126500 = 2.2.3.3.3.3.5.5.5.7.13.40343 ... I have found the minimal k value such that k.p+1 is prime for the first 12 primes p . Here it is : k=2258581791060=2.2.3.3.5.7.7.971.263723

Here is my approach of this problem :

a) k is an even number .Indeed if k is odd , 3.k+1 is even.
b) k=0 mod 3. Indeed if k=1 mod 3 , 2.k+1=0 mod 3, if k=2 mod 3 , 7*k+1=0 mod 3

c) k=0 mod 5 . Indeed if k=1 mod 5 , 19*k+1=0 mod 5

if k=2 mod 5 , 2*k+1=0 mod 5
if k=3 mod 5 , 3*k+1=0 mod 5
if k=4 mod 5 , 11*k+1=0 mod 5

d) k=0 mod 7 . Indeed if k=1 mod 7 , 13*k+1=0 mod 7

if k=2 mod 7 , 3*k+1=0 mod 7
if k=3 mod 7 , 2*k+1=0 mod 7
if k=4 mod 7 , 5*k+1=0 mod 7
if k=5 mod 7 , 11*k+1=0 mod 7
if k=6 mod 7 , 29*k+1=0 mod 7

Therefore k is of the form : k=210*m.

Over more for the first 10 primes , we have :m=0;1;6 or 8 mod 11 the first 11 primes, we have : m=0;1 or 8 mod 11 the first 12 primes , we have : m=0 or 1 mod 11

Wilke wrote

The smallest value I've found which satisfies the conditions of the problem is 37202202450; i.e. 37202202450*k + 1 is prime for k = 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Let P denote the product of 2k + 1, 3k + 1, 5k + 1, 7k + 1, 11k + 1, 13k + 1, 17k + 1, 19k + 1, 23k + 1 and 29k + 1. Then considering the residues of the individual factors (mod r) for r = 2, 3, 5 and 7, it is easy to show that the product is divisible by 2, 3, 5 or 7 unless k is a multiple of 2*3*5*7 = 210. Then one can easily set up code which tries multiples of 210 as possible values of k and generates the product P. Then if

gcd(P,6469693230) > 1, we proceed to the next value of k. (6469693230 is the product of the first 30 primes.)

Then one sets up a sieve for each factor using a Fermat theorem test. The value of k = 210*177153345 = 37202202450. The corresponding values of each factor of P are 74404404901, 111606607351, 186011012251, 260415417151, 409224226951, 483628631851, 632437441651, 706841846551, 855650656351 and 1078863871051. That each of these is prime was verified using UBASIC's ecmx program.

***

Warf found by his own calculations the asked solution k= 37202202450 "...It was a numerical search but taking advantage of the same sieving that Rosa described."

***

Phil Carmody wrote:

"Next 3 terms 148684126500 and 2258581791060 and 161082438032880 [8/3/04]"

Later - the same day - he added:

"and another, number 14: 24581646307811670 "

Then almost one month later, he wrote again:

"I accidentally left my program running for several weeks (maybe a month!) I was only searching for 14s, but found 3 15s in the process!

Here are all the 14s:

24581646307811670
39055674088756710
79565092682014170
187231486701238980
262260412776497520
472549586664565410
656422023176535000
732677307101562900
1095883279219325520
1183192161007235610
1229573338316071980
1264128917439090780
1280383876469553240
1581963212913969270
1652784273771770370
1782603781183482180
1844573648776946010
1931039388156287640
1953226765141191540
2187367341150106500
2226258759701725980
2449882138257641010
2520231288741469380
2626881197880169230
2779573811003028270
2791365043355648760
2810623488909071220
3345387534651517740
3570758835454976550
4002123460460010270
4139477891763459630
4252710523844502720
4330249453053989850
4641402144052983540
4745139032813999820
4946661784515169950
5459821650103770180
5498746144270604610
5531940030180466980
6008290332226475370
6044330330247476190
6158034781993229520
7382603195510539920
7437473314587686370
7463014630493124780
8452739497646953740
8465448552769897920
8686873456895340630
9311435976194437470
9488427197189043180
9993949639276952580
10055870246010922410
11873020883253483030
12062272620328140210
12272321084015751990

of which

2791365043355648760
1183192161007235610
8686873456895340630

are 15s (primes up to k=47)

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