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 Integers inside surds (Posted on 2019-02-03)
Find all integers N for which the expression

√(N + 70710678√173) - √(N - 70710678√173)

has an integer value.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating

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 computer doesn't mind tedium, and discovers duplicates Comment 2 of 2 |
(In reply to Easy Peasy by FrankM)

While there are 126 values for N+P and corresponding N-P, there are only 63 values for N, as discovered by sorting the output from a program that evaluates N (lexicographic order rather than numeric order, but it accomplishes the pairing):

1011142649466
1011142649466
112351084314
112351084314
1181130214
1181130214
1215445126
1215445126
1249999995805094
1249999995805094
1255048326
1255048326
138888888424326
138888888424326
14044943788486
14044943788486
14697886188454
14697886188454
15432098727654
15432098727654
1560549446694
1560549446694
157809727334
157809727334
1633098596166
1633098596166
165146097926
165146097926
1714677760966
1714677760966
173395614726
173395614726
17546594886
17546594886
181456576614
181456576614
18361206054
18361206054
19277265254
19277265254
1972101094
1972101094
20172435206
20172435206
2059248294
2059248294
2144890566
2144890566
216249999274251334
216249999274251334
24027777697139046
24027777697139046
2429775272744486
2429775272744486
2669753077459974
2669753077459974
269975030305734
269975030305734
27300845769094
27300845769094
28327262586
28327262586
296639230829606
296639230829606
29997225596646
29997225596646
3033427378086
3033427378086
3215349594
3215349594
3333025130374
3333025130374
337048120134
337048120134
37455494246
37455494246
416666665268826
416666665268826
46296296145594
46296296145594
4681647970554
4681647970554
4899295435386
4899295435386
5144032946586
5144032946586
520183518426
520183518426
52606896666
52606896666
544366551834
544366551834
55052191194
55052191194
57801863994
57801863994
5881752954
5881752954
60488703546
60488703546
6151828986
6151828986
72083333091417114
72083333091417114
8009259232379706
8009259232379706
809925090915066
809925090915066
84961424966
84961424966
889917692486874
889917692486874
89991676770714
89991676770714
9100281944154
9100281944154
9462783654
9462783654
968144826
968144826
982408026
982408026
997806234
997806234
9999075218106
9999075218106

being the sorted output of

4 kill "intinsrd.txt"
5 open "intinsrd.txt" for output as #2
10 prd = 173 * 2^2 * 3^6 * 89^2 * 14713^2
20  for a=0 to 1
30  for b=0 to 6
40 for c=0 to 2
50 for d=0 to 2
60      npp=173^a * 2 * 3^b * 89^c * 14713^d
70      nmp=prd/npp
80      n = (npp+nmp)/2
90      print #2, int(n):print int(n)
100 next
110 next
120 next
130 next
140 close #2

 Posted by Charlie on 2019-02-06 12:56:35

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