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I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

A video on this topic, titled "Researchers thought this was a bug," is on the 3Blue1Brown YouTube channel here.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

A video on this topic, titled "Researchers thought this was a bug," is on the 3Blue1Brown YouTube channel here.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

A video on this topic, titled "Researchers thought this was a bug," is on the 3Blue1Brown YouTube channel here.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.

Define $\mathop{\mathrm{sinc}} x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \mathop{\mathrm{sinc}} x = \pi/2$

They then found the following results:

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3)= \pi/2$

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5) \ldots \mathop{\mathrm{sinc}} (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5) \ldots \mathop{\mathrm{sinc}} (x/15)=$ $467807924713440738696537864469\pi/935615849440640907310521750000$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package: $\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.