Problems & Puzzles: Puzzles

Puzzle 872. Integers (p^2)*Q Emmanuel Vantieghem sent the following nice puzzle: The last two weeks I used to browse through my old notes. One of them could result in a "puzzle" or a "problem" for your site.   It is about numbers of the form  (p^2)*Q, where  Q  is squarefree and  p  a prime bigger than the greatest prime factor of  Q. Here are the first such numbers : 4, 9, 18, 25, 49, 50, 75, 98, 121, 147, 150, 169, 242, 245, 289, 294, 338, 361, 363, 490, ... (A229027  at the OEIS) I was mainly interested in consecutive integers of that form. It was easy to find many examples.  These are the smallest ones :                18490, 18491                23762, 23763                36517, 36518                   ... I believe that there are infinitely many of them.  So, this can be the first question :   Q1.Are there infinitely many consecutive numbers in  A229027 ?  (It would already be fine if you can send huge numbers).   A more difficult question is to find three consecutive integers of that form.  There are none below 2*10^10.  So, this could give    Q2. Find three consecutive integers in that sequence or prove that this is impossible.

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Contributions came from J. K. Andersen, Jaroslaw Wreblowski and Emmanuel Vantieghem

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Andersen wrote:

Q2
1375618025554869 = 3 * 31 * 37 * 47 * 347 * 4951^2
1375618025554870 = 2 * 5 * 151 * 839 * 983 * 1051^2
1375618025554871 = 7 * 149 * 449 * 677 * 2083^2

It may not be the smallest triple.

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Jarek Wreblowski wrote:

Regarding Q1:

Suppose p, 2p-1, 8p-3 and 8p-1 are primes (it is reasonable to
conjecture that there are infinitely many such quads of prime numbers,
but there is no hope for a proof).
Then the numbers
n=(2p-1)*(8p-1)^2
and
n+1=2*p*(8p-3)^2
are consecutive and satisfy the conditions of the puzzle (their prime
factorizations are as above).
An example is
p=10^333+6669479355069
which leads to n>10^1000.

On the other hand my exhaustive search indicates that there are 23785
pairs of consecutive numbers below 2*10^13 satisfying conditions of
the puzzle. You can find the list of the smaller numbers of each pair
here:
http://www.math.uni.wroc.pl/~p-k1g4/Puzzle872.txt

Regarding Q2:

These 23785 registers also answers to Q2, as there is a unique triple of consecutive
numbers below 2*10^13 satisfying the puzzle, namely the one starting
from 16036630184409.

A selective search produced many more such triples (the smallest
numbers of each triple are listed):

1229115033835845
1375618025554869
2800845811122985
3765423670679405
43776182208766793
947248470852685053
1369606836889235129
2820054728028623145
4917319920246921097
7349269937628887741
9283652432561414461
16536057526702187929
21403316798587146981
22241305575094178065
26669248562500846201
27653646203380869341
28073902394832124053
48296730966636634689
66877326781285958065
73884570129365166253
112466228761639028913
258303591567135983289
353870638459033075753
605731475529172816185
614148748615485268205
707494482829007492945
727947061097721948581
1304796244244073311569
1323816659739916295933
1816454500872460744033
2411663939571334949213
3293717286435782402181
4370940091726584223801
4656348944164188087313
5953400431042621684837
6521667921398572346733
9302678566562181969433
12265759597530744294329
14608912647204327285733
15769196159390791930813
27525763857231152410605
41136447389945332046809
43454192542791693863069
46097420758674162951117
59355140868035541674965
60395682750523578939817
60917555593362067814829
68718437977018626888265
72033149302146625287341
111929082515508880570045
119922378805311611045341
161515406009117565234569
189644662759804031685009
191424483845396641914769
204714472374870395005209
205307949569109352552389
207139926175664897873621
273803333599405939139237
295033668330731078436665
303770491460420114910593
484082831302841673930165
513124766261796910026765
824484625634162849879913
967438348326356319193269
1186720320634985511638537
1245466521449526869602961
1635125542824258517904537
1947526348933707751096413
2096762085769509821035253
4825141718263930267648717
5179131814305116849762369
14336717011162478519291129
21820829842324463305757717

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Emmanuel wrote:

I still cannot find a triple of consecutive numbers on that kind.
 

But it is rather easy to find big examples of two consecutive numbers.

My champion is :
 

x = 479161859630796025043369447794764241475230310528782289365446
= 2*3*282595665109120179111713717329^2
 

x+1 =11*1735997*158405513301027124152253979^2

 

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