# [seqfan] Re: [math-fun] Triangular+Triangular = Factorial

Richard Guy rkg at cpsc.ucalgary.ca
Tue Sep 14 18:11:09 CEST 2010

```Many thanks for all that help.  Just to clear up
one or two things that might be in doubt:

1.  I omitted the solution (x,y,z) = (89,269,8).

2.  For  z = 59, there are 32 solutions, which
some hero(ine) may like to calculate from

? factor(8*59!+2)
%64 =
[2 1]

[8249057 1]

[2174500369 1]

[67820061593 1]

[114446404287889 1]

[11247609217977437 1]

[354241345536913447681853 1]

3. There are no solutions for  z = 69, as may
be seen from

? factor(4*69!+1)
%74 =
[96493309986243088621721365030167853723206604771 1]

[7093650428305879089240130905502588632647558330525131 1]

4.  The first values of  z  about which I am ignorant
are  z = 71, 74, 80, 81, 93, 99, 100, ...

5.  If one is willing to accept probable primes,
then the following sequence of  z  for which
4 * z!  +  1   is prime may be complete for
z < 300 or more?? ---  comments?  [not in OEIS]

0, 1, 2, 4, 7, 8, 9, 13, 16, 28, 54, 129, 190, ...

R.

On Mon, 13 Sep 2010, Georgi Guninski wrote:

> On Mon, Sep 13, 2010 at 09:35:14AM +0800, Douglas McNeil wrote:
>> and can fully factor 8*(z!)+2 for
>>
>> fully factored: [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16, 17, 21, 24,
>> 27, 28, 29, 32, 33, 34, 42, 49, 54, 59, 66, 68, 72, 79, 85, 86, 95,
>> 96, 102, 118, 129, 135, 164, 184, 190, 219, 221, 264, 351, 357, 457,
>> 466]
>>
>> This leaves only 7 numbers, [69, 74, 80, 81, 93, 99, 100] <= 100 as
>> undecided (123 in total < 500) but they could be within reach.  I set
>> a pretty short time limit on the ecm runs, to get a first pass done
>> overnight.  Anyway, it should be complete up to z <= 68.
>>
>> Note that there are two differences with results in this thread.
>> Trivial: 2 I think was put in by accident in the previous exclusion
>> list (8*(2!)+2 = 18 = 9+9); I agree with everything else.
>> Non-trivial: the results quoted by R.K. Guy listed no solution for
>> z=59, but 8*(59!)+2 has prime factors
>>
>
> thanks for catching the multiplicity bug with 2, hopefully fixed now.
>
> i can add 139 as fully factored solution (factorization at end):
>
> and 294,286 as no solutions.
>
> here are my results for z<=300
>
> 42 solutions:
>
> 2,3,4,5,7,8,9,10,13,15,16,17,21,24,27,28,29,32,33,34,42,49,54,59,66,72,79,85,86,95,96,102,118,129,135,139,164,184,190,219,221,264
>
> 52 with unknown status:
>
> 61, 68, 69, 71, 74, 80, 81, 93, 99, 100, 107, 109, 114, 125, 126, 131, 133, 134, 140, 141, 143, 162, 165, 171, 173, 178, 189, 192, 199, 208, 209, 212, 217, 220, 222, 224, 233, 234, 235, 242, 244, 248, 252, 254, 259, 268, 275, 277, 280, 291, 296, 299
>
> and the rest are without solutions.
>
> took about 6 hours with opportunistic EC factoring.
>
> all known to me <=300 (E&OE) , format is |SOLUTION(+NUMBER OF PRIME
> FACTORS)|
>
> 2(+2),3(+2),4(+2),5(+3),6(-),7(+2),8(+2),9(+2),10(+3),11(-),12(-),13(+2),14(-),15(+3),16(+2),17(+3),18(-),19(-),20(-),21(+3),22(-),23(-),24(+3),25(-),26(-),27(+5),28(+2),29(+3),30(-),31(-),32(+4),33(+4),34(+6),35(-),36(-),37(-),38(-),39(-),40(-),41(-),42(+3),43(-),44(-),45(-),46(-),47(-),48(-),49(+6),50(-),51(-),52(-),53(-),54(+2),55(-),56(-),57(-),58(-),59(+7),60(-),62(-),63(-),64(-),65(-),66(+3),67(-),70(-),72(+3),73(-),75(-),76(-),77(-),78(-),79(+6),82(-),83(-),84(-),85(+3),86(+2),87(-),88(-),89(-),90(-),91(-),92(-),94(-),95(+4),96(+3),97(-),98(-),101(-),102(+5),103(-),104(-),105(-),106(-),108(-),110(-),111(-),112(-),113(-),115(-),116(-),117(-),118(+6),119(-),120(-),121(-),122(-),123(-),124(-),127(-),128(-),129(+2),130(-),132(-),135(+5),136(-),137(-),138(-),139(+4),142(-),144(-),145(-),146(-),147(-),148(-),149(-),150(-),151(-),152(-),153(-),154(-),155(-),156(-),157(-),158(-),159(-),160(-),161(-),163(-),164(+4),166(-),167(-),168(-),169(-),170(-),172(-),174(-),175(-),176(-
),
> 177(-),179(-),180(-),181(-),182(-),183(-),184(+5),185(-),186(-),187(-),188(-),190(+2),191(-),193(-),194(-),195(-),196(-),197(-),198(-),200(-),201(-),202(-),203(-),204(-),205(-),206(-),207(-),210(-),211(-),213(-),214(-),215(-),216(-),218(-),219(+5),221(+3),223(-),225(-),226(-),227(-),228(-),229(-),230(-),231(-),232(-),236(-),237(-),238(-),239(-),240(-),241(-),243(-),245(-),246(-),247(-),249(-),250(-),251(-),253(-),255(-),256(-),257(-),258(-),260(-),261(-),262(-),263(-),264(+4),265(-),266(-),267(-),269(-),270(-),271(-),272(-),273(-),274(-),276(-),278(-),279(-),281(-),282(-),283(-),284(-),285(-),286(-),287(-),288(-),289(-),290(-),292(-),293(-),294(-),295(-),297(-),298(-),300(-)
>
>
> 139+(4): [2, 288245637707785632653, 16377513261243153166696673, 81476289222097686957291918164413236312063831684966837348397616965036974678013728645423662413401987536260023582443949960880989765406242254362489665549605338264289123236751948012164984705741862629]
>
> if someone is interested in the (partial) factorizations, let me know.
>
>
>
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>
>
>

```