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Why isn't the last statement simplified? Is it a bug in 11.1.1.0?

Product[1 - 1/Prime[n]^2, {n, 1, Infinity}]
FullSimplify[(1 - 1/Prime[n]) (1 + 1/Prime[n])]
Product[%, {n, 1, Infinity}]
Product[(1 - 1/Prime[n]) (1 + 1/Prime[n]), {n, 1, Infinity}]
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    $\begingroup$ I wouldn't consider the fact that Mathematica can't compute everything that can be computed a bug ... Functions like Product use heuristics. It's not uncommon that helping them along a bit allows finding a solution they can't get on their own. (That said, it may be worth reporting to WRI. Only the developers will be able to tell if this behaviour is as it should be, or whether this is an easy opportunity for improvement.) $\endgroup$ – Szabolcs Aug 23 '17 at 10:57
  • $\begingroup$ @Szabolcs, I usually give hints as needed. Make your comment an answer and I will sign off on it. Thanks. $\endgroup$ – Fred Kline Aug 23 '17 at 11:01
  • $\begingroup$ I also found that the first-time evaluate of that last expression takes about 10s (with the kernel already running so it's not startup time, and subsequent evaluations take ~10^-4 s). Curious as to what it's doing during that time. $\endgroup$ – aardvark2012 Aug 23 '17 at 11:15
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    $\begingroup$ It isn't exactly a bug, so if you do report, please mention that this is a suggestion. $\endgroup$ – J. M. will be back soon Aug 23 '17 at 11:21
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    $\begingroup$ @J.M., From WRI support: I have filed your suggestion with our developers. Thank you for taking the time to report it. $\endgroup$ – Fred Kline Aug 25 '17 at 3:16
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Copied my last comment to create an answer to close this thread.

From WRI support: I have filed your suggestion with our developers. Thank you for taking the time to report it.

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