(1) a+b>rad(ab(a+b))
where rad is the radical, the product of all unique prime factors of a number. I.e. rad(4)=2, rad(6)=6, rad(60)=30, rad(81)=3, rad(101)=101, ...
There is an infinite number of pairs (a,b) as (1), but given a real number epsilon>0 there seems to be only a finite number of abc-triplets (a,b,a+b) such that
(2) a+b>rad(ab(a+b))^(1+epsilon)
and that's one version of the famous abc-conjecture.
: abcpair \ a b -- flag
locals| b a |
a b ugcd 1 =
a b +
dup a ugcd 1 =
swap b ugcd 1 =
and and ;
test if (a,b,a+b) is a abc-triplet, and
: unusual \ a b -- flag
: unusual \ a b -- flag
locals| b a |
a b 2dup + * * radical
a b + < ;
test if a+b>rad(ab(a+b)).
1 1000 condition non create-set zdup cardinality . 999 ok
creates the set {1,...,999}.
zdup cartprod zdup cardinality . 998001 ok
create the Cartesian product {1,...,999}x{1,...,999} and
2dim < filter-set zdup cardinality . 498501 ok
filter the set so that the first component is less than the second.
2dim abcpair filter-set zdup cardinality . 303791 ok
This skip all (a,b) but those where (a,b,a+b) is a abc-triplet.
2dim unusual filter-set zdup cardinality . 32 ok
This is the remaining set of pairs such that (1):
zdup cr zet.
{(1,8),(1,48),(1,63),(1,80),(1,224),(1,242),(1,288),(1,512),(1,624),(1,675),(1,728),(1,960),(2,243),(3,125),(4,121),(5,27),(5,507),(7,243),(13,243),(25,704),(27,512),(32,49),(32,343),(49,576),(81,175),(81,544),(100,243),(104,625),(169,343),(200,529),(343,625),(640,729)} ok
Considering the pairs as Gaussian integers and transform the set of unusual pairs to their Gaussian norms give:
zdup 2dim gnorm transform-set zdup cardinality . cr zet. 32
{65,754,2305,3425,3970,6401,14657,15634,37186,50177,58565,59053,59098,59218,69049,82945,118673,146210,257074,262145,262873,302497,319841,334177,389377,401441,455626,496241,508274,529985,921601,941041} ok
Since the both sets have 32 elements I hasten to raise the conjecture:
(3) All (ordered) unusual pairs has unique Gaussian norms.
To test the conjecture for different limits without stack overflow, conj3 works:
2dup abcpair 0=
if 2drop false
else unusual
then ;
true locals| flag |
0 >zst \ empty set on zst stack
2
?do i 1
?do i j abcunusual
if i j gnorm dup zdup smember
if false to flag drop i j pad 2! leave
else >zst zfence union
then
then
loop flag 0= if leave then
loop flag ;
100 conj3 . -1 ok
zet. {65,754,2305,3425,3970,6401} ok
5000 conj3 . -1 ok
zdup cardinality . 87 ok
cr zet.
{65,754,2305,3425,3970,6401,14657,15634,37186,50177,58565,59053,59098,59218,69049,82945,118673,146210,257074,262145,262873,302497,319841,334177,389377,401441,455626,496241,508274,529985,921601,941041,1048577,1048601,1242793,1476226,1569061,1750393,1837097,2566561,2944705,3067769,3317074,4093154,4101154,4194385,4213625,4484017,4584929,4735097,4783069,4798594,4916545,5303810,5592434,5646001,5760001,5774602,5831545,8977273,8998393,9144577,9439993,9765746,9976306,10185529,11944561,12379505,13302409,13986466,14548594,15108770,15745025,15784466,15818497,15944098,16769026,16777337,16778441,17155426,18548777,19131877,23070401,23660897,23819585,24153953,33667138} ok
True so far. This take some time but I try 10000:
10000 conj3 . -1 ok
zdup cardinality . 129 ok
cr zet.
{65,754,2305,3425,3970,6401,14657,15634,37186,50177,58565,59053,59098,59218,69049,82945,118673,146210,257074,262145,262873,302497,319841,334177,389377,401441,455626,496241,508274,529985,921601,941041,1048577,1048601,1242793,1476226,1569061,1750393,1837097,2566561,2944705,3067769,3317074,4093154,4101154,4194385,4213625,4484017,4584929,4735097,4783069,4798594,4916545,5303810,5592434,5646001,5760001,5774602,5831545,8977273,8998393,9144577,9439993,9765746,9976306,10185529,11944561,12379505,13302409,13986466,14548594,15108770,15745025,15784466,15818497,15944098,16769026,16777337,16778441,17155426,18548777,19131877,23070401,23660897,23819585,24153953,26040898,31640674,31706945,32283521,33667138,34000562,37515986,37520281,38950162,39052481,39421505,40947202,40985921,43033601,43050817,43445377,44289026,44446210,47045882,47046137,47048690,56811506,64000361,64235537,65713618,66928882,66961570,68374489,68508353,70761674,73260281,73530626,85470281,87890626,88510465,93655426,94008377,96040001,96060226,118771553,123820633,140639489,141533305} ok
(To be continued)
Have to college student joggers or maybe young people along with finding out ailments get hold of exceptional rentals for college or university? The reason or perhaps have you thought to? What's rational? https://imgur.com/a/kC06Mig https://imgur.com/a/vL2s8fs https://imgur.com/a/En7OV9g https://imgur.com/a/FwvHSSu https://imgur.com/a/DnEkYrt https://imgur.com/a/UNXrNce https://imgur.com/a/IxGUHal
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