Problems & Puzzles: Puzzles

 Puzzle 650 Triangular, Primes & Powers of 2 JM Bergot sent the following puzzle: Take the triangular number 45 45-43=2 45-41=4 45-37=8 45-29=16 45-13=32 Bergot ask for triangular numbers T(n) such that T(n)-2^i is a prime for i=1 to k and composite for i=k+1. I found that T(65)=2145 is such that 2145-2^i is prime for i=1 to 8 and composite for i=9. Q1. Can you provide a better example? Q2. What if the powers of 2 need not to be consecutive?

Contributions came from Torbjörn Alm, Jud McCranie, W. Edwin Clark, Ryan Bailey

Torbjörn wrote:

Q1:  No better solutions found below T(1000000000) = 5*10^17, only a handful of size=8

Q2: One better solution:

Solution length 10   n= 85892925  T(n)= 3688797325474275

3688797325474275-3688797325474273=2
3688797325474275-3688797325474271=4
3688797325474275-3688797325474267=8
3688797325474275-3688797325474259=16
3688797325474275-3688797325474243=32
3688797325474275-3688797325474211=64
3688797325474275-3688797325474147=128
3688797325474275-3688797325474019=256  non-prime
3688797325474275-3688797325473763=512
3688797325474275-3688797325473251=1024

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Jud wrote:

Q1. I got to wondering if there was some reason that you couldn't have more than 8 in a row, but I could not find a reason why not.  I checked for n to over 2,000,000,000 and didn't find any sequences longer than 8.

Q2. I got these records:

5                    15    3
6                    21    4
9                    45    5
14                   105    6
30                   465    7
65                  2145    9
114                  6555   10
870                378885   11
1550               1202025   12
3254               5295885   13
18690             174667395   15
51650            1333887075   16
324414           52622383905   18
651629          212310502635   19
7342290        26954614893195   20
10347609        53536511182245   21
67804649      2298735246908925   22

Notice that except for n=6 the triangular numbers are all multiples of 15.  This makes the triangular number minus a power of 2 more likely to be prime since they aren't divisible by 3 or 5.

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W.E. Clark wrote:

Q1 The triangular number T(13744665) = 94457914853445 can be written in
the form prime + 2^j for the  9 consecutive numbers j in { 32, 33, 34, 35, 36, 37, 38, 39, 40} [CR, Bravo!]

Q2 For the triangular number T(22906334) = 262350080112945, I find that  it can be written in the form prime + 2^j for the 21  values of j in the set
{1, 2, 4, 6, 12, 14, 15, 20, 21, 22, 26, 27, 30, 32, 35, 36, 38, 39, 42, 44, 45}

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Ryan wrote:

Q1. I am convinced no better example exists after checking all triangular numbers between 1 and 1 X 10^14. For prime numbers to have  gaps of 2, 4, 8, 16, 32, 64, 128, and 256 respectively would be near impossible. [CR, See result for Q1 from WEC above]

Q2. There are many solutions with nonconsecutive i's. My largest yields a total i count of 23.

51423374463534255
51423374463534255 - 4 = 51423374463534251 with i = 2
51423374463534255 - 16 = 51423374463534239 with i = 4
51423374463534255 - 1024 = 51423374463533231 with i = 10
51423374463534255 - 2048 = 51423374463532207 with i = 11
51423374463534255 - 4096 = 51423374463530159 with i = 12
51423374463534255 - 32768 = 51423374463501487 with i = 15
51423374463534255 - 65536 = 51423374463468719 with i = 16
51423374463534255 - 131072 = 51423374463403183 with i = 17
51423374463534255 - 1048576 = 51423374462485679 with i = 20
51423374463534255 - 2097152 = 51423374461437103 with i = 21
51423374463534255 - 4194304 = 51423374459339951 with i = 22
51423374463534255 - 33554432 = 51423374429979823 with i = 25
51423374463534255 - 268435456 = 51423374195098799 with i = 28
51423374463534255 - 1073741824 = 51423373389792431 with i = 30
51423374463534255 - 2147483648 = 51423372316050607 with i = 31
51423374463534255 - 4294967296 = 51423370168566959 with i = 32
51423374463534255 - 34359738368 = 51423340103795887 with i = 35
51423374463534255 - 68719476736 = 51423305744057519 with i = 36
51423374463534255 - 137438953472 = 51423237024580783 with i = 37
51423374463534255 - 8796093022208 = 51414578370512047 with i = 43
51423374463534255 - 17592186044416 = 51405782277489839 with i = 44
51423374463534255 - 35184372088832 = 51388190091445423 with i = 45
51423374463534255 - 140737488355328 = 51282636975178927 with i = 47
With 51423374463534255 - 281474976710656 = 51141899486823599 being composite for i = 48
t = 320697285

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